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P & C (A) Arrangement 1. An English school and a Vernacular school are both under one superintendent. Suppose that the superintendentship, the four different teachership of English school and Vernacular school each, are vacant, if there be altogether 11 candidates for the appointments, 3 of whom apply exclusively for the superintendentship and 2 exclusively for the appointment in the English school, the number of ways in which the different appointments can be disposed of is: [3, –1] (A) 4320 (B) 268 (C) 1080 (D*) 25920 2. How many positive integers > 9 have their digits strictly increasing from left to right. Ans. 502 [4, 0] 3. Number of permutations of the word " AUROBIND " in which vowels appear in an alphabetical order is : [5, –1] (A*) P (8, 4) (B) C (8, 4) (C*) 4 ! C (8, 4) (D) C (8, 5) . 5 ! [ Hint : A, I, O, U treat them alike . Now find the arrangement of 8 letters in which 4 alike and 4 different = 8 ! ] 4 ! 4. Sum of all the numbers that can be formed using all the digits 2, 3, 3, 4, 4, 4 is: (A*) 22222200 (B) 11111100 (C) 55555500 (D) 20333280 5. Delegates from 9 countries includes countries A, B, C, D are to be seated in a row. The number of possible seating arrangements, when the delegates of the countries A and B are to be seated next to each other and the delegates of the countries C and D are not to be seated next to each other is : (A) 10080 (B) 5040 (C) 3360 (D*) 60480 6. Number of ways in which 7 people can occupy six seats, 3 seats on each side in a first class railway compartment if two specified persons are to be always included and occupy adjacent seats on the same side, is (k). 5 ! then k has the value equal to: (A) 2 (B) 4 (C*) 8 (D) none 7. A gentleman invites a party of m + n (m n) friends to a dinner & places m at one table and n at another, the table being round. If the clockwise & anticlockwise arrangements are not to be distinguished and assuming sufficient space on both tables, then the number of ways in which he can arrange the guest is (A) (m n)! 4mn (B*) 1 2 (m n)! 4mn (C) 2 (m n)! 4mn (D) none 8. Five identical balls are to be distributed among 10 identical boxes. If not more than one ball goes into a box, the total number of ways this can be done is _. Ans : 252 9. There are m identical white & n identical black balls with m > n . The number of different ways in which all the balls are put in a row so that no black balls are side by side is : (A) m ! (m1) ! (m n 1) ! (B) (m n1) ! (m 1) ! n ! (C*) (m1) ! n ! (m n 1) ! (D) none 10. How many different four digit numbers be formed using digits 0, 1, 2, 3, 4, 5 if each number should contain 1 (repitition of digits is not allowed) (A) 108 (B) 180 (C*) 204 (D) none of these 11. It a rule (say) in Scotland that consonant (s) cannot be placed between a strong and a weak vowel. The strong vowels are a, 0, u and the weak vowels are e & i. Show that the whole number of words that can be

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P & C (A) Arrangement 1. An English school and a Vernacular school are both under one superintendent. Suppose that the superintendentship, the four different teachership of English school and Vernacular school each, are vacant, if there be altogether 11 candidates for the appointments, 3 of whom apply exclusively for the superintendentship and 2 exclusively for the appointment in the English school, the number of ways in which the different appointments can be disposed of is: [3, –1] (A) 4320 (B) 268 (C) 1080 (D*) 25920 2. How many positive integers > 9 have their digits strictly increasing from left to right. Ans. 502 [4, 0] 3. Number of permutations of the word " AUROBIND " in which vowels appear in an alphabetical order is : [5, –1] (A*) P (8, 4) (B) C (8, 4) (C*) 4 ! C (8, 4) (D) C (8, 5) . 5 ! [ Hint : A, I, O, U treat them alike . Now find the arrangement of 8 letters in which 4 alike and 4 different = 8 ! ] 4 ! 4. Sum of all the numbers that can be formed using all the digits 2, 3, 3, 4, 4, 4 is: (A*) 22222200 (B) 11111100 (C) 55555500 (D) 20333280 5. Delegates from 9 countries includes countries A, B, C, D are to be seated in a row. The number of possible seating arrangements, when the delegates of the countries A and B are to be seated next to each other and the delegates of the countries C and D are not to be seated next to each other is : (A) 10080 (B) 5040 (C) 3360 (D*) 60480 6. Number of ways in which 7 people can occupy six seats, 3 seats on each side in a first class railway compartment if two specified persons are to be always included and occupy adjacent seats on the same side, is (k). 5 ! then k has the value equal to: (A) 2 (B) 4 (C*) 8 (D) none 7. A gentleman invites a party of m + n (m n) friends to a dinner & places m at one table and n at another, the table being round. If the clockwise & anticlockwise arrangements are not to be distinguished and assuming sufficient space on both tables, then the number of ways in which he can arrange the guest is (A) (m n)! 4mn (B*) 1 2 (m n)! 4mn (C) 2 (m n)! 4mn (D) none 8. Five identical balls are to be distributed among 10 identical boxes. If not more than one ball goes into a box, the total number of ways this can be done is _. Ans : 252 9. There are m identical white & n identical black balls with m > n . The number of different ways in which all the balls are put in a row so that no black balls are side by side is : (A) m ! (m1) ! (m n 1) ! (B) (m n1) ! (m 1) ! n ! (C*) (m1) ! n ! (m n 1) ! (D) none 10. How many different four digit numbers be formed using digits 0, 1, 2, 3, 4, 5 if each number should contain 1 (repitition of digits is not allowed) (A) 108 (B) 180 (C*) 204 (D) none of these 11. It a rule (say) in Scotland that consonant (s) cannot be placed between a strong and a weak vowel. The strong vowels are a, 0, u and the weak vowels are e & i. Show that the whole number of words that can be

- 1. P & C (A) Arrangement 1. An English school and a Vernacular school are both under one superintendent. Suppose that the superintendentship, the four different teachership of English school and Vernacular school each, are vacant, if there be altogether 11 candidates for the appointments, 3 of whom apply exclusively for the superintendentship and 2 exclusively for the appointment in the English school, the number of ways in which the different appointments can be disposed of is: [3, –1] (A) 4320 (B) 268 (C) 1080 (D*) 25920 2. How many positive integers > 9 have their digits strictly increasing from left to right. Ans. 502 [4, 0] 3. Number of permutations of the word " AUROBIND " in which vowels appear in an alphabetical order is : [5, –1] (A*) P (8, 4) (B) C (8, 4) (C*) 4 ! C (8, 4) (D) C (8, 5) . 5 ! [ Hint : A, I, O, U treat them alike . Now find the arrangement of 8 letters in which 4 alike and 4 different = 8 4 ! ! ] 4. Sum of all the numbers that can be formed using all the digits 2, 3, 3, 4, 4, 4 is: (A*) 22222200 (B) 11111100 (C) 55555500 (D) 20333280 5. Delegates from 9 countries includes countries A, B, C, D are to be seated in a row. The number of possible seating arrangements, when the delegates of the countries A and B are to be seated next to each other and the delegates of the countries C and D are not to be seated next to each other is : (A) 10080 (B) 5040 (C) 3360 (D*) 60480 6. Number of ways in which 7 people can occupy six seats, 3 seats on each side in a first class railway compartment if two specified persons are to be always included and occupy adjacent seats on the same side, is (k). 5 ! then k has the value equal to: (A) 2 (B) 4 (C*) 8 (D) none 7. A gentleman invites a party of m + n (m n) friends to a dinner & places m at one table and n at another, the table being round. If the clockwise & anticlockwise arrangements are not to be distinguished and assuming sufficient space on both tables, then the number of ways in which he can arrange the guest is (A) mn 4 )! n m ( (B*) 2 1 mn 4 )! n m ( (C) 2 mn 4 )! n m ( (D) none 8. Five identical balls are to be distributed among 10 identical boxes. If not more than one ball goes into a box, the total number of ways this can be done is ___________. Ans : 252 9. There are m identical white & n identical black balls with m > n . The number of different ways in which all the balls are put in a row so that no black balls are side by side is : (A) m m m n ! ( ) ! ( ) ! 1 1 (B) ( ) ! ( ) ! ! m n m n 1 1 (C*) ( ) ! ! ( ) ! m n m n 1 1 (D) none 10. How many different four digit numbers be formed using digits 0, 1, 2, 3, 4, 5 if each number should contain 1 (repitition of digits is not allowed) (A) 108 (B) 180 (C*) 204 (D) none of these
- 2. 11. It a rule (say) in Scotland that consonant (s) cannot be placed between a strong and a weak vowel. The strong vowels are a, 0, u and the weak vowels are e & i. Show that the whole number of words that can be formed with this condition of (n + 3) letters each formed of 'n' consonants & the vowels a, e, o is ) 2 n ( ! ) 3 n ( 2 . (Assume no repetition of a letter) 12. The number of permutations which can be formed out of the letters of the word "SERIES" taking three letters together is: (A) 120 (B) 60 (C*) 42 (D) none 13. The number of ways in which a mixed double tennis game can be arranged from amongst 9 married couple if no husband & wife plays in the same game is: (A) 756 (B) 3024 (C*) 1512 (D) 6048 14. The number of different words of three letters which can be formed from the word "PROPOSAL", if a vowel is always in the middle are : [3, –1] (A*) 53 (B) 52 (C) 63 (D) 32 15. A shop sells 6 different flavours of ice - cream. In how many ways can a customer choose 4 ice - cream cones if [6, 0] (i) they are all of different flavours (ii) they are not all of different flavours (iii) they contain exactly 3 different flavours (iv) they contain only 2 or 3 different flavours ? Ans. (i) 15 (ii) 111 (iii) 60 (iv) 105 16. A man is dealt a poker hand (consisting of 5 cards) from an ordinary pack of 52 playing cards. The number of ways in which he can be dealt a "straight" (a straight is five consecutive values not of the same suit, eg. {Ace, 2, 3, 4, 5}, {2, 3, 4, 5, 6}.......................... & {10, J, Q, K, Ace}) is (*A) 10 (45 4) (B) 4 !. 210 (C) 10. 210 (D*) 10200 [5, –1] 17. 18 guests have to be seated, half on each side of a long table. 4 particular guests desire to sit on 1 particular side & 3 others on the other side. Determine the number of ways in which the sitting arrangement can be made. Ans. 5 11 C (9 !)2 18. Number of natural number between 100 and 1000 such that at least one of their digits is 6, is (A) 243 (B*) 252 (C) 258 (D) 648 19. In the decimal system of numeration the number of 6-digit numbers in which the sum of the digits is divisible by 5 is- (A*) 180000 (B) 540000 (C) 5 × 105 (D) none of these 20. There are 6 roads b/w A & B & 4 Roads b/w B & C Find: (i) No. ways in which you can drive from A B C (ii) In how many ways you can drive A B C then return back also C B A (iii) In how many ways you can drive A – B – C & back without using same road again. Ans. (i) 24 (ii) 576 (iii) 360 21. How car number plates can be formed if each plate consists of 2 Alphabets, 1 consonant & 1 vowel & then followed by 3 different digits from (0, 1......, 9) Ans. 26 × 5 × 10 × 9 × 8
- 3. 22. If Repetition is not permitted then: (i) How many 3 digit number can be formed with digits {2, 3, 5, 6, 7, 9} (ii) How many of these are less than 400 (iii) How many of them are odd Ans. (i) 120 (ii) 40 (iii) 80 23. Let Pm stand for m Pm . Then the expression 1. P1 + 2. P2 + 3. P3 +...... + n. Pn = (A*) (n + 1) ! 1 (B) (n + 1) ! + 1 (C) (n+ 1)! (D) none [ Hint: Tn = n n Pn = n. n ! = n ! ((n + 1) 1) = Tn = (n + 1) ! n ! Now put n = 1, 2, 3,...... n and add ] 24. There are 12 balls numbered from 1 to 12. The number of ways in which they can be used to fill 8 places in a row so that balls are numbered in either ascending or descending order is: (A) 12 C8 (B) 12 P8 (C) 2 × 12 P8 (D*) 2 × 12 C8 25. How many words can be formed using all the letter of the word TENDULKAR if the words should start with a vowel & end with a consonant. Ans. 6. 3. 7! 26. 6 Boys & 4 Girls enter a Railway compartment having 5 seats on each side. In how many ways can they occupy seats if the Girls are to occupy corner seats: (A*) 17280 (B) 12780 (C) 5400 (D) 720 27. Number of numbers divisible by 25 that can be formed using only the digits 1, 2, 3, 4, 5 & 0 taken five at a time is: (A) 2 (B) 32 (C*) 42 (D) 52 [ Hint: 1, 2, 3, 4, 5, 0 A number divisible by 25 if the last two digits are 25 or 50 Hence if 5 is not taken then number of numbers = 0 if 2 is not taken then 5 0 = 6 then in each case 5 0 = 6 we have 10 numbers 2 5 = 4 2. 2 1 If 0 is not taken 2 5 = 6 Hence total = 30 + 6 + 6 = 42 28. 4 normal distinguishable dice are rolled once. The number of possible outcomes in which atleast one die shows up 2 is: (A) 216 (B) 648 (C) 625 (D*) 671 [ Hint: Total no dice shows up 2 = 64 54 = 671 ] 29. Delegates from 9 countries includes countries A, B, C, D are to be seated in a row. The number of possible seating arrangements, when the delegates of the countries A and B are to be seated next to each other and the delegates of the countries C and D are not to be seated next to each other is: (A) 10080 (B) 5040 (C) 3360 (D*) 60480 [ Hint: I H G F E AB 7 C2 2 ! 6 ! 2 ! = 60480 ] 30. The number of numbers that can be formed by using digits 3, 4, 5, 6, 5, 4, 3 so that the odd digits always occupy the odd places is: (A) 7 2 3 ! ( !) (B) 4!. 3! (C*) 4 3 2 3 ! . ! ( !) (D) none 31. Number of permutations of the word " AUROBIND" in which vowels appear in an alphabetical order is: (A*) P (8, 4) (B) C (8, 4) (C*) 4 ! C (8, 4) (D) C (8, 5). 5 !
- 4. [ Hint: A, I, O, U treat them alike. Now find the arrangement of 8 letters in which 4 alike and 4 different = 8 4 ! ! ] 32. Number of three digits even number which can be formed with the condition that if 5 is one of the digid, then 7 is the next digit is: (A) 5 (B) 325 (C) 345 (D*) 365 33. There are 10 seats in a double decker bus, 6 in the lower deck and 4 on the upper deck. Ten passengers board the bus, of them 3 refuse to go to the upper deck and 2 insist on going up. The number of ways in which the passengers can be accommodated is _____. [ Ans.: 4 C2 . 2! 6 C3 . 3! 5! or 172800 ] 34. Number of ways in which the letters of the word "ORION" can be arranged if the two consonants are not adjacent in any arrangement is ______. [Ans. 5 2 4 2 ! ! ! 2 ! = 36 ] 35. The number of three digit numbers having only two consecutive digits identical is: (A) 153 (B*) 162 (C) 180 (D) 161 [ Hint: x x when two consecutive digits are 11, 22, etc = 9. 9 = 81 0 0 when two consecutive digits are 0 0 = 9 x x when two consecutive digits are 11, 22, 33,... = 9. 8 = 72 Total] 36. There are unlimited number of identical balls of 4 different colours. Number of arrangements of atmost 8 balls in a row which can be made by using them is ______. [ Ans. 4 + 42 + 43 +.... + 48 = 4 3 ( 48 1) = 87380 ] 37. Six cards are drawn one by one from a set of unlimited number of cards, each card is marked with numbers 1, 0 or 1. Number of different ways in which they can be drawn if the sum of the numbers shown by them vanishes, is: (A) 111 (B) 121 (C*) 141 (D) none [ Hint: 0 0 0 0 0 0 1 way; 0 0 0 0 1, 1 6 4 ! ! = 30 ways; 0 0 1 1 1 1 6 2 2 2 ! ! ! ! = 90 ways; 1 1 1 1 1 1 6 3 3 ! ! ! = 20 ways Total = 141 Alternatively co-efficient of x0 in (1 + x 1 + x2 )6 ] 38. The number of arrangements that can be made taking 4 letters, at a time, out of the letters of the word " PASSPORT " is: (A*) 606 (B) 4464 (C) 4356 (D) 4644 39. Find the number of 7 lettered words each consisting of 3 vowels and 4 consonants which can be formed using the letters of the word "DIFFERENTIATION". Ans. 532770 40. The number of 5 digit numbers of the form x y z y x in which x < y is: (A) 350 (B*) 360 (C) 380 (D) 390 [ Hint: The first digit ' x ' can be any one of from 1 to 8 where as ' z ' can be any one from 0 to 9 when x = 1, y = 2, 3, 4,......, 9 x = 2, y = 3, 4,......, 9 and so on Thus the total = (8 + 7 + 6 +...... + 2 + 1) 10 = 360 ]
- 5. 41. Number of proper divisors of 2520 which are divisible by 10 is ______ & the sum of these divisors is ______. [ Ans. 17, 4760 ] 42. If as many more words as possible be formed out of the letters of the word "UNIQUE" then the number of words in which the relative order of vowels and consonants remain unchanged is ________. [ Hint: x x x x = 4 2 ! ! . 2 ! 1 = 23 ] 43. A forecast is to be made of the results of five cricket matches, each of which can be a win or a draw or a loss for Indian team. Let p = number of forecasts with exactly 1 error q = number of forecasts with exactly 3 errors and r = number of forecasts with all five errors then the incorrect statement is: (A) 2 q = 5 r (B) 8 p = q (C*) 8 p = 5 r (D) 2 (p + r) > q [ p = 10; q = 80; r = 32 ] 44. Number of ways in which the letters of the word " DEEPMALA " can be arranged on a circle, if A' s are always together but E's are to be separated distinguishing between the clockwise and the anticlockwise arrangement, is: (A) 360 (B*) 240 (C) 120 (D) none of these [ Hint: D P M L A A = D M P L X say 4 !. 5 C2 = 240 ] 45. In a chess tournament, where the participants are to play one game with one another two players fell ill, having played only 3 games each. If the total number of games played in the tournament is equal to 84 then the number of participants in the beginning was ______ [ Ans.:15; Hint: n 2 C2 + 6 = 84 n = 15 or n = 10 (rejected) if the two players have also played then n 2 C2 + 5 = 84 No solution ] 46. Number of even integers from 1000 and 9999 (both inclusive) having all distinct digits is ______. [ Ans.: 2296 ] [ Hint: When ' 0 ' occupies the units place = 9. 8. 7 0 = 504 when 2, 4, 6 or 8 occupies the units place Number of ways of filling units, 4th, 3rd & 2nd place is 4, 8, 8 & 7 = 4 8 8 7 = 1792 Total = 2296 ] 47. There are counters available in 3 different colours (atleast four of each colour). Counters are all alike except for the colour. If ' m ' denotes the number of arrangements of four counters if no arrangement consists of counters of same colour and ' n ' denotes the corresponding figure when every arrangement consists of counters of each colour, then: (A) m = 2 n (B*) 6 m = 13 n (C) 3 m = 5 n (D) 5 m = 3 n [ Hint: m = 34 3 = 78 n = 34 3 2 2 3 4 = 81 45 = 36 Hence m n = 78 36 = 13 6 6 m = 13 n B ] 48. Set A consists of 4 distinct elements & the set B consists of 5 distinct elements. Number of mapping defined from A B which are not injective is: (A) 600 (B) 550 (C*) 505 (D) none [ 54 5! = 505 ] 49. Team A & B play in a tournament. The first team that wins two games in a row or wins a total o f four games is considered to win the tournament. The number of ways in which tournament can occur is _______. [ Ans. 14 ]
- 6. 50. Number of all 5 digit whole numbers formed on the screen of a calculator which can be recognised as 5 digit numbers with perfectly normal digits when they are read inverted, is _______. [ 0 1 2 5 6 8 9:: 6.7.7.7.6 = 12348 ] 51. Total number of ways in which the number 510510 can be resolved as a product of two factors which are relatively prime, is _______. [ Ans 64 ] [ Hint: 510510 = 2. 3. 5. 7. 11. 13. 17 ] 52. Let Pn denotes the number of ways of selecting 3 people out of ' n ' sitting in a row, if no two of them are consecutive and Qn is the corresponding figure when they are in a circle. If Pn Qn = 6, then ' n ' is equal to: (A) 8 (B) 9 (C*) 10 (D) 12 [ Hint: Pn = n 2 C3 ; Qn = n C3 [ n + n (n 4) ] or Qn = n n C C 1 4 2 3 . Pn Qn = 6 n = 10 ] 53. For some natural N, the number of positive integral ' x ' satisfying the equation, 1 ! + 2 ! + 3 ! +...... + (x !) = (N)2 is: (A) none (B) one (C*) two (D) infinite [ x = 1 & x = 3 ] 54. Number of increasing permutations of ' m ' symbols are there from the ' n ' set of numbers { a1 , a2 ,......, an } where the order among the numbers is given by, a1 < a2 < a3 <...... < an is ______________. [ Ans.: n Cm ] 55. The maximum number of permutations of 2n letters in which there are only a's & b's, taken all at a time is given by: (A*) 2n Cn (B*) 2 1 6 2 10 3 4 6 1 4 2 . . ...... . n n n n (C*) n n n n n n n n 1 1 2 2 3 3 4 4 2 1 1 2 . . . ...... . (D*) 2 1 3 5 2 3 2 1 n n n n . . . ...... ( ) ( ) ! 56. The smallest positive integer n with 24 divisors (including 1 and n) is __________. [ Ans. 360] [ Hint: The number of divisors of N = pa qb rc sd ... where p, q, r, s are distinct and a, b, c, d N is (a + 1) (b + 1) (c + 1)...... A number with 24 divisors must be of the form of 223 ; 211 . 3; 27 . 32 ; 22 . 31 . 51 . 71 ; 23 . 32 . 5 smallest is 360 and the next higher is 420 ] 57. There are counters available in x different colours. The counters are all alike except for the colour. The total number of arrangements consisting of y counters, assuming sufficient number of counters of each colour, if no arrangement consists of all counters of the same colour is: (A*) xy x (B) xy y (C) yx x (D) yx y 58. Let there be n 3 circles in a plane. The value of n for which the number of radical centres, is equal to the number of radical axes is: ( Assume that all radical axes and radical centre exist and are different) (A) 7 (B) 6 (C*) 5 (D) none [ Hint: n C2 = n C3 b = 5 ]
- 7. 59. A set 'A' consists of 6 & the set B consists of 3 distinct elements. The total number of functions from A B which are surjective is ______. [ Hint: required number = total functions number of functions which are into = 36 [3. (26 2) + 3] = 36 [186 + 3] = 36 189 = 540. Note that surjective means onto ] 60. Number of natural numbers of not more than twenty digits, which can be formed using the digits 0, 1, 2, 3 & 4, is ______. [ Ans. 520 1 ] [ Hint: (5x5x........x5) 20 times (when '0' occupies all the places] 61. Number of natural numbers less than 1000 and divisible by 5 can be formed with the ten digits, each digit not occuring more than once in each number is ______. [ Hint: single digit = 1; two digit = 9 + 8 = 17; three digit = 72 + 64 = 136 Total = 154 ] 62. A letter lock consists of three rings each marked with fifteen different letters. It is found that a man could open the lock only after he makes half the number of possible unsuccessful attempts to open the lock. If each attempt takes 10 secs. the time he must have spent is not less than: (A*) 4 1 2 hours (B) 5 1 2 hours (C) 6 1 4 hours (D) 9 hours 63. On the normal chess board as shown, I1 & I2 are two insects which starts moving towards each other. Each insect moving with the same constant speed. Insect I1 can move only to the right or upward along the lines while the insect I2 can move only to the left or downward along the lines of the chess board. The total number of ways the two insects can meet at same point during their trip is: (A*) 9 8 10 7 11 6 12 5 13 4 14 3 15 2 16 1 (B*) 28 1 1 3 2 5 3 7 4 9 5 11 6 13 7 15 8 (C*) 2 1 6 2 10 3 14 4 18 5 22 6 26 7 30 8 (D*) C (16, 8) [ Hint:(8 C0 . 8 C0 ) + (8 C1 . 8 C1 )+. ...+ (8 C8 . 8 C8 ) = 16 C8 ) = 12870 ] 64. Messages are conveyed by arranging 4 white, 1 blue and 3 red flags on a pole. Flags of the same colour are alike. If a message is transmitted by the order in which the colours are arranged then the total number of messages that can be transmitted if exactly 6 flags are used is: (A) 45 (B) 65 (C) 125 (D*) 185 [ Hint: consider, 4 alike + 2 others alike, 4 A + 2 different 3 A + 3 OA and 3 A + 2 OA + 1 different 15 + 30 + 20 + 120 ] 65. Number of different words that can be formed using all letters of the word "ELEVEN" if each word neither begins nor ends in E, is: (A*) 24 (B) 36 (C) 48 (D) none [ Hint: n A B = 36; n (A B) = 24; n B A = 36; n A B = 24 ] 66. A five letter word is to be formed such that the letters appearing in the odd numbered positions are taken from the letters which appear without repetition in the word "MATHEMATICS". Further the letters appearing in the even numbered positions are taken from the letters which appear with repetition in the same word "MATHEMATICS". The number of ways in which the five letter word can be formed is: (A) 720 (B*) 540 (C) 360 (D) none [ Hint: H E C I S; M A T; x x x 5 C3 . 3 ! (3 C1 + 3 C2 . 2 !) = 5 P3 . 9 = 540 ]
- 8. 67. Three digit numbers in which the middle one is a perfect square are formed using the digits 1 to 9. Their sum is: (A*) 134055 (B) 270540 (C) 170055 (D) none of these [ Hint: Middle place 1, 4 & 9 Two terminal positions 1, 2,......, 9 Hence total numbers = 9. 9. 3 = 243 For the middle place 1, 4 & 9 will come 81 times sum = 81 10 (1 + 4 + 9) A For units place each digit from 1 to 9 will appear 27 times sum = 27 )1 + 2 +...... + 9) B For hundreath's place, similarly sum = 27 10 (1 + 2 +...... + 9) C A + B + C gives the required sum ] 68. n objects are arranged in a row. A subset of these objects is called unfriendtly, if no two of its elements are consecutive. Show that the number of unfriendly subsets of a k element set is k n k 1 . Sol. There are (n k + 1) spaces between the crosses, and we have to take 'k' out of it. 1, 2, 3,......., n 0 0 0 , , ......... k elements n k + 1 ck x x x x x n k remaining ...... ( ) ] 69. The number of other ways the letters of the word "HONOLULU" can be arranged taken all at a time is: (A) 5040 (B*) 5039 (C) 2 7 ! (D) none 70. The number of 6 letter word that can be formed out of the letters of the word "ASSIST" in which the S's come alternate with other letters is: (A*) 12 (B) 24 (C) 36 (D) none 71. The number of numbers that can be formed by using digits 3, 4, 5, 6, 5, 4, 3 so that the odd digits always occupy the odd places is: (A) 7 2 3 ! ( !) (B) 4!. 3! (C*) 4 3 2 3 ! . ! ( !) (D) none 72. The number of ways in which the letters of the word "CONSTANT" can be arranged without changing the relative positions of the vowels & consonants is: (A*) 360 (B) 256 (C) 444 (D) none 73. Number of 5 digit number which can be read in the same way from the left & from the right is ______. [ Hint: 1st place 9 ways; 2nd and 3rd places in10 ways; 4th and 5th must be the same digit as 2nd and 1st 9 · 10 · 10 = 900 ] 74. Number of numbers greater than a million and divisible by 5 which can be formed by using only the digits 1, 2, 1, 2, 0, 5 & 2 is: (A) 120 (B*) 110 (C) 90 (D) none [ Hint: 0 = 6 2 3 1 ! ! ! ! = 60 5 = 6 2 3 1 ! ! ! ! 5 2 3 ! ! ! = 50 60 + 50 = 110 ] 75. There are m apples and n oranges to be placed in a line such that the two extreme fruits being both oranges. Let P denotes the numbe of arrangements if the fruits of the same species are different and Q the corresponding figure when the fruits of the same species are alike, then the ratio P/Q has the value equal to: (A*) n P2 . m Pm . (n 2)! (B) m P2 . n Pn . (n 2)! (C) n P2 . n Pn . (m 2)! (D) none
- 9. [ Hint: P = n C2 . 2 ! (m + n 2) ! Q = m n n m 2 2 ! ! ! ] 76. There are 'm' men & 'n' monkeys (n > m). Then match the entries of column I & II. Column I Column II (A) Number of ways in which each man may (i) nm become the owner of one monkey is (ii) n Pm (B) Number of ways in which every monkey (iii) mn has a master, if a man may have any number (iv) mn of monkeys is: [ Ans. n Cm . m!; (m. m. m..... n times) A (ii), B (iv) ] 77. 10 different letters of an alphabet are given. Words with 5 letters are formed from these given letters. Then the number of words which have atleast one letter repeated is: (A*) 69760 (B) 30240 (C) 99748 (D) none [ Hint: Total number of words with all different letters = 105 10 P5 = 69760 ] 78. The number of permutations of 4 letters that can be made out of the letters of the word "EXAMINATION" is: (A) 1896 (B) 2136 (C*) 2454 (D) none 79. The number of three digit numbers having only two consecutive digits identical is: (A) 153 (B*) 162 (C) 180 (D) 161 [ Hint: x x when two consecutive digits are 11, 22, etc = 9. 9 = 81 0 0 when two consecutive digits are 0 0 = 9 x x when two consecutive digits are 11, 22, 33,... = 9. 8 = 72 Total] 80. The streets of a city are arranged like the lines of a chess board. There are m streets running North to South & 'n' streets running East to West. The number of ways in which a man can travel from NW to SE corner going the shortest possible distance is: (A) m n 2 2 (B) ( ) . ( ) m n 1 1 2 2 (C) ( ) ! ! . ! m n m n (D*) ( ) ! ( ) ! . ( ) ! m n m n 2 1 1 [ Hint: (m 1) paths of one kind & (n 1) paths of other kind, taken all at a time ] 81. How many three-digit numbers can be formed without using the digits 0, 2, 3, 4, 5 and 6? [Ans. 64] 82. Two cards are drawn one at a time & without replacement from a pack of 52 card. The number of ways in which the two cards can be drawn, are (A*) 2652 (B) 2704 (C) 2500 (D) none of these 83. If repetitions are not allowed (i) How many 3-digit numbers can be formed from the six digits 2, 3, 5, 6, 7 and 9. [Ans. 120] (ii) How many of these are less than 400? [Ans. 40] (iii) How many are even? [Ans. 40] (iv) How many are odd? [Ans. 80] (v) How many are multiples of 5? [Ans. 20] 84. How many numbers divisible by 5 and lying between 4000 and 5000 can be formed from the digits 4, 5, 6, 7 and 8. [Ans. 25] 85. The digits, from 0 to 9 are written on slips of paper and placed in a box. Three of the slips of paper are drawn and placed in order. How many different outcomes are possible? (A) 1000 (B*) 720 (C) 810 (D) none of these
- 10. 86. 5 boys & 3 girls are sitting in a row of 8 seats. Number of ways in which they can be seated so that not all the girls sit side by side is (A*) 36000 (B) 9080 (C) 3960 (D) 11600 87. Number of 6 digit numbers which can be formed using digits 1, 2, 3, 4, 5, 6 & 7, if each digit is to be used at most once & terminal digits are to be even, is........ [Ans. 720] 88. The number of signals that can be made with 3 flags each of different colour by hoisting 1 or 2 or 3 above the other is (A) 3 (B) 7 (C*) 15 (D) 16 89. The number of numbers from 1000 to 9999 (both inclusive) that do not have all 4 different digits is (A) 4048 (B*) 4464 (C) 4518 (D) 4536 90. The number of seven digit numbers that can be written using only three digits 1, 2 & 3 under the condition that the digit 2 occurs exactly twice in each number is: (A*) 672 (B) 640 (C) 512 (D) none of these 91. In a conference 10 speakers are present. If S1 wants to speak before S2 & S2 wants to speak after S3, then the number of ways all the 10 speakers can give their speeches with the above restriction if the remaining seven speakers have to objection to speak at any number is: (A) 10C3 (B) 10P8 (C) 10P3 (D*) 3 ! 10 92. How many four digit numbers are there all whose digits are odd, if repetition of digits is allowed. [Ans. 625] 93. How many numbers greater than 23000 can be formed from the digits 1, 2, 3, 4, 5? [Ans. 90] 94. In how many ways can the letters of the word ‘ARRANGE’ be arranged so that (i) The two R’S are never together (ii) The two A’S are together but not two R’S. (iii) Neither two A’S nor the two R’S are together. [Ans. 900, 240, 660] 95. Number of ways in which 7 people can occupy six seats, 3 seats on each side in a first class railway compartment if two specified persons are to be always included and occupy adjacent seat on the same side is (K). 5!, then K has the value equal to (A) 2 (B) 4 (C*) 8 (D) none of these 96. How many different nine digits numbers can be formed from the number 223355888 be rearranging the digits so that the odd digits occupy even positions? (A) 16 (B) 36 (C*) 60 (D) 180 97. How many different words can be formed out of the letters of the word ‘ALLAHABAD’? In how many of them the vowels occupy the even positions? [Ans. 7560, 60] 98. The total number of ways in which six ‘+’ and four ‘–’ signs can be arranged in a line such that no two ‘–’ signs occur together is............... [Ans. 35] 99. Every body in a room shakes hands with everybody else. The total number of hand shakes is 66. The total number of persons in the room is............ [Ans. 12] 100. The number of ways of arranging of the letters AAAAA, BBB, CCC, D, EE and F in a row if the letter C are separated from one another is
- 11. (A*) ! 2 ! 3 ! 5 ! 12 . C3 13 (B) ! 2 ! 3 ! 3 ! 5 ! 13 (C) ! 2 ! 3 ! 3 ! 14 (D*) ! 6 ! 13 . 11 101. A tea party is arranged for 16 people along two sides of a long table with 8 chairs on each side. Four men wish to sit on one particular side and two on the other side. In how many ways can they be seated? [Ans. 10!. 30. 82. 72] 102. The tamer of wild animals has to bring one by one 5 lions & 4 tigers to the circus arena. The number of ways this can be done if no two tigers immediately follow each other is........ [Ans. 43200] 103. In how many ways can the letters of the word ‘CINEMA’ be arranged so that order of vowels do not change. [Ans. 120] 104. An eight-oared boat is to be manned by a crew chosen from 11 men, of whom 3 can steer but cannot row, and the rest can row but cannot steer. In how many ways can the crew be arranged, If two of the man can only row on bow side? [Ans. 25920] 105. Numbers of natural numbers smaller than ten thousand and divisible by 4 using the digits 0, 1, 2, 3 and 5 without repetition is: (A) 18 (B) 27 (C) 32 (D*) 31 106. There are ‘m’ points on a straight line AB & n points on the line AC none of them being the point A. Triangles are formed with these points as vertices when (i) A is excluded (ii) A is included. The ratio of numbers of triangles in the two cases is. (A*) n m 2 n m (B) 1 n m 2 n m (C) 1 n m 2 n m (D) ) 1 n )( 1 m ( ) 1 n ( m 107. The total number of ways in which 8 men & 6 women can be arranged in a line so that no 2 women are together is: (A) 48 (B*) 8P8. 9P6 (C) 8! (84) (D) 8C8. 9C8 108. The number of ways in which 8 non-identical apples can be distributed among 3 boys such that every boy should get at least 1 apple & atmost 4 apples is (k. 7P3) where k has the values: (A) 88 (B) 66 (C) 44 (D*) 22 109. The number of different ways in which five ‘dashes’ and eight ‘dots’ can be arranged, using only seven of these 13 ‘dashes’ & ‘dots’ is (A) 1287 (B) 119 (C*) 120 (D) 1235520 110. 10 IIT & 2 PET students sit in a row. If the number of ways in which exactly 3 IIT students sit between 2 PET students is K.10!, then the value of ‘K’ is: (A*) 16. 10! (B) 2.10! (C) 12! (D) none of these 111. 3 different railway passes are allotted to 5. The Number of ways in which it can be done is - (A*) 60 (B) 20 (C) 15 (D) 10 112. The number of three digit numbers having only two consecutive digits identical is: (A) 153 (B*) 162 (C) 180 (D) 161 [ Hint: x x when two consecutive digits are 11, 22, etc = 9. 9 = 81 0 0 when two consecutive digits are 0 0 = 9 x x when two consecutive digits are 11, 22, 33,... = 9. 8 = 72 Total] 113. The number of different seven digit numbers that can be written using only three digits 1, 2 & 3 under the condition that the digit 2 occurs exactly twice in each number is: (A*) 672 (B) 640 (C) 512 (D) none Hint: 7 C2 . 25
- 12. 114. Sum of all four digit numbers formed using only the digits 2, 2, 3, 1 is - (A) 26644 (B*) 26664 (C) 39996 (D) 53328 115. In a shooting competition a man can score 0, 2 or 4 points for each shot. Then the number of different ways in which he can score 14 points in 5 shots, is: (A) 20 (B) 24 (C*) 30 (D) none [ Ans.: 30 [ Hint: 44420 or 44222 ! 2 ! 3 ! 5 ! 3 ! 5 = 30 ways ] 116. Number of different natural numbers which are smaller than two hundred millions & using only the digits 1 or 2 is: (A) 504 (B*) 766 (C) 804 (D) none of these 117. Let Pm stand for m Pm . Then the expression 1 . P1 + 2 . P2 + 3 . P3 + ...... + n . Pn = (A*) (n + 1) ! 1 (B) (n + 1) ! + 1 (C) (n + 1) ! (D) none [ Hint : Tn = n n Pn = n . n ! = n ! ((n + 1) 1) = Tn = (n + 1) ! n ! Now put n = 1, 2, 3, ...... n and add ] 118. The value of the sum 1995 P1 + 1995 2 2 P ! + 1995 3 3 P ! + ...... + 1995 1995 1995 P ( )! = ______ . [ Ans. (21995 1) ] 119. There are 12 balls numbered from 1 to 12. The number of ways in which they can be used to fill 8 places to a Row so that Balls are numbered in Ascending of descending order is (A) 12 C8 (B) 12 P8 (C) 2 × 12 P8 (D*) 2 × 12 C8 120. How many words can be formed using all the letter of the word TENDULKAR if : (i) the words should start with a vowel AND end with a consonant . (ii) the words should start with a vowel OR end with a consonant . [ Ans. : (i) 6 3 7 !(ii) 6 8 ! + 3 . 8 ! 6 . 8 . 7 ! ] 121. 6 Boys & 4 Girls enter a Railway compartment having 5 seats on each side. In how many ways can they occupy seats in the Girls are to occupy corner seats - (A*) 17280 (B) 12780 (C) 17289 (D) None 122. There are m identical white & n identical black balls with m > n . The number of different ways in which all the balls are put in a row so that no black balls are side by side is : (A) m m m n ! ( ) ! ( ) ! 1 1 (B) ( ) ! ( ) ! ! m n m n 1 1 (C*) ( ) ! ! ( ) ! m n m n 1 1 (D) none 123. 4 normal distinguishable dice are rolled once . The number of possible outcomes in which atleast one die shows up 2 is : (A) 216 (B) 648 (C) 625 (D*) 671 [ Hint : Total no dice shows up 2 = 64 54 = 671 ] 124. The number of other ways the letters of the word "HONOLULU" can be arranged taken all at a time is : (A) 5040 (B*) 5039 (C) 2 7 ! (D) none 125. The number of numbers that can be formed by using digits 3, 4, 5, 6, 5, 4, 3 so that the odd digits always occupy the odd places is : (A) 7 2 3 ! ( !) (B) 4 ! . 3 ! (C*) 4 3 2 3 ! . ! ( !) (D) none 126. Number of permutations of the word " AUROBIND" in which vowels appear in an alphabetical order is (A*) P (8, 4) (B) C (8, 4) (C*) 4 ! C (8, 4) (D) C (8, 5) . 5 ! [ Hint : A, I, O, U treat them alike . Now find the arrangement of 8 letters in which 4 alike
- 13. and 4 different = 8 4 ! ! ] 127. The number of ways in which the letters of the word "CONSTANT" can be arranged without changing the relative positions of the vowels & consonants is : (A*) 360 (B) 256 (C) 444 (D) none 128. Number of ways in which the letters of the word "ORION" can be arranged if the two consonants are not adjacent in any arrangement is ______ . [ 5 2 4 2 ! ! ! 2 ! = 36 ] 129. Number of numbers greater than a million and divisible by 5 which can be formed by using only the digits 1, 2, 1, 2, 0, 5 & 2 is : (A) 120 (B*) 110 (C) 90 (D) none [ Hint : 0 = 6 2 3 1 ! ! ! ! = 60 5 = 6 2 3 1 ! ! ! ! 5 2 3 ! ! ! = 50 60 + 50 = 110 ] 130. There are m apples and n oranges to be placed in a line such that the two extreme fruits being both oranges . Let P denotes the numbe of arrangements if the fruits of the same species are different and Q the corresponding figure when the fruits of the same species are alike, then the ratio P/Q has the value equal to : (A*) n P2 . m Pm . (n 2) ! (B) m P2 . n Pn . (n 2) ! (C) n P2 . n Pn . (m 2) ! (D) none [ Hint : P = n C2 . 2 ! (m + n 2) ! Q = m n n m 2 2 ! ! ! ] 131. The result of 21 football matches (win, lose or draw) are to be predicted . The number of different forecasts which contain exactly 18 correct results is : (A) 21 C3 (B) 3 . 21 C3 (C*) 8 . 21 C3 (D) 27 . 21 C3 [ Hint : 21 C18 . 23 = 21 C3 . 8 ] 134. Sum of all the numbers that can be formed using all the digits 2, 3, 3, 4, 4, 4 is : (A*) 22222200 (B) 11111100 (C) 55555500 (D) 20333280 [ Hint : 4 x 30 [x] + 3 x 20 [x] + 2 x 10 [x] where [x] = 1 + 10 + 102 + 103 + 104 + 105 ] 135. There are (p + q) different books on different topics in Mathematics. (p q) If L = The number of ways in which these books are distributed between two students X and Y such that X get p books and Y gets q books. M =The number of ways in which these books are distributed between two students X and Y such that one of them gets p books and another gets q books. N = The number of ways in which these books are divided into two groups of p books and q books then, (A) L = M = N (B) L = 2M = 2N (C*) 2L = M = 2N (D) L = M = 2N 136. Number of n digit numbers which consists of the digits 1 & 2 only if each digit is to be used atleast once, is : (A) 2n - 1 2 (B) 2n (C) 2n 1 (D*) 2n 2 [Hint: (2 x 2 x .............2) n times-(when 1 or 2 is there at all the n places][ Ans. 2n 2 ] 137. If 'D' denotes the number of ways in which 8 different things be distributed between Ram and Shyam and 'L' denotes the corresponding figure when the things are all alike, each person receiving atleast one thing in both the cases then, D L = ______ . [ Ans. : 254 7 = 247 ; Hint : (28 2) 7 ]
- 14. 138. There are 'm' men & 'n' monkeys (n > m) . Then match the entries of column I & II . Column I Column II (A) Number of ways in which each man may (i) nm become the owner of one monkey is (ii) n Pm (B) Number of ways in which every monkey (iii) mn has a master, if a man may have any number (iv) mn of monkeys is : [ Ans. n Cm . m! ; (m . m . m ..... n times) A (ii) , B (iv) ] 139. 10 different letters of an alphabet are given . Words with 5 letters are formed from these given letters . Then the number of words which have atleast one letter repeated is : (A*) 69760 (B) 30240 (C) 99748 (D) none [ Hint : Total number of words with all different letters = 105 10 P5 = 69760 ] 140. The number of permutations of 4 letters that can be made out of the letters of the word "EXAMINATION" is : (A) 1896 (B) 2136 (C*) 2454 (D) none 141. Number of numbers greater than 1000 which can be formed using only the digits 1, 1, 2, 3, 4, 0 taken four at a time is : (A) 112 (B) 123 (C) 332 (D*) 159 [ Hint : 2 alike + zero + 1 different = 27 ; 2 alike + 2 non zero + different = 36 ; All four different '0' always included = 72 ; All four different non zero digit = 24 159] 142. The number of three digit numbers having only two consecutive digits identical is : (A) 153 (B*) 162 (C) 180 (D) 161 [ Hint : x x when two consecutive digits are 11, 22, etc = 9 . 9 = 81 0 0 when two consecutive digits are 0 0 = 9 x x when two consecutive digits are 11, 22, 33, ... = 9 . 8 = 72 Total] 143. Number of ways in which n students can be partitioned into two teams containing atleast one student is : (A) 2n 2 (B) 2n 2 (C*) 2n 1 1 (D) None 144. There are unlimited number of identical balls of 4 different colours . Number of arrangements of atmost 8 balls in a row which can be made by using them is ______ . [ Ans. 4 + 42 + 43 + .... + 48 = 4 3 ( 48 1) = 87380 ] 145. Number of proper divisors of 2520 which are divisible by 10 is ______ & the sum of these divisors is ______ . [ Ans. 17 , 4760 ] 146. The streets of a city are arranged like the lines of a chess board . There are m streets running North to South & 'n' streets running East to West . The number of ways in which a man can travel from NW to SE corner going the shortest possible distance is : (A) m n 2 2 (B) ( ) . ( ) m n 1 1 2 2 (C) ( ) ! ! . ! m n m n (D*) ( ) ! ( ) ! . ( ) ! m n m n 2 1 1 [ Hint : (m 1) paths of one kind & (n 1) paths of other kind, taken all at a time ] 147. Sum of all the digits used in writting all the numbers from 1 to 1000 is : (A) 12741 (B) 13946 (C) 4996 (D*) 13501 [ Hint : S = 0 + 1 + 2 + 3 + ...... + 998 + 999 S = 999 + 998 + ........ + 1 + 0 2 S = 999 + 999 + .... (1000 times) (sum of the digits being 27 in each case) = 1000 27 S = 13501 ] 148. The number of ways in which 3 letters can be mailed if 4 letter boxes are available is : (A) 4 P3 (B) 4 C3 (C) 34 (D*) 43
- 15. 149. A five letter word is to be formed such that the letters appearing in the odd numbered positions are taken from the letters which appear without repetition in the word "MATHEMATICS" . Further the letters appearing in the even numbered positions are taken from the letters which appear with repetition in the same word "MATHEMATICS". The number of ways in which the five letter word can be formed is : (A) 720 (B*) 540 (C) 360 (D) none [ Hint : H E C I S ; M A T ; x x x 5 C3 . 3 ! (3 C1 + 3 C2 . 2 !) = 5 P3 . 9 = 540 ] 150. Number of three digits even number with the condition that if 5 is one of the digit, then 7 is the next digit is, _______ [ Ans. 5 + 8.9.5 = 365 ] 151. Find the number of natural numbers smaller than 10000 & divisible by 4 which can be formed using the digits 0, 1, 2, 3 & 5 if repetition of digits is not allowed. [ Ans. 31 ] 152. Number of natural numbers of not more than twenty digits, which can be formed using the digits 0, 1, 2, 3 & 4, is ______ . [ Hint : (5 5 ........ 5) 20 times (when '0' occupies all the places] [ Ans. 520 1 ] 153. The results of 11 chess matches (as win, lose or draw) are to be forecast . Out of all possible forecasts, find the number of ways in which 8 correct & 3 incorrect results can be forecasted . [ Ans. : 11 C8 8 ] 154. Number of numbers from 1000 to 10000 which have none of their digits repeated are ______ & ______ of these are odd . [ Ans. 4536 & 2240 ] [ Hint : Total = 9 . 9 . 8 . 7 = 4536 number of odd numbers = x = 8 . 8 . 7 . 7 = 2240 ] 155. Find the number of 6 digit numbers which can be formed using the digits 1, 2, 3, 4, 5, 6 & 7, if each digit is to be used atmost once & terminal digits are to be even . [ Ans. 720 = 3 P2 . 5 P4 ] 156. Seven speakers P1 , P2 , P3 , ......, P7 must address the conference and P2 must not exceed P1 . Find the number of ways to establish the successive order of speaches . [ Ans: 2520 ] 157. Number of proper divisors of 2520 which are divisible by 10 is ______ & the sum of these divisors is ______ . [ Ans. 17 , 4760 ] 158. A set 'A' consists of 6 & the set B consists of 3 distinct elements . The total number of functions from A B which are surjective is ______ . [ Hint : required number = total functions number of functions which are into = 36 [3 . (26 2) + 3] = 36 [186 + 3] = 36 189 = 540 , note surjective means onto] 159. The number of numbers that can be formed by using digits 3 , 4 , 5 , 6 , 5 , 4 , 3 so that the odd digits always occupy the odd places is : (A) 7 2 3 ! ( !) (B) 4 ! . 3 ! (C*) 4 3 2 3 ! . ! ( !) (D) none 160. Number of ways in which the letters of the word RESONANCE can be arranged such that , vowels occupy odd places is : (A*) 3600 (B) 720 (C) 600 (D) none of these 161. A man is at the origin on the xaxis & takes a unit step either to the left or to the right . He stops after 5 steps or if he reaches 3 or 2 . Number of ways in which he : (A*) reaches 2 is 3 (B) reaches 3 is 4 (C) stop exactly after walking 5 steps is 12 (D*) can perform the experiment is 20 [ for C the correct is 16 ] 162. Number of numbers greater than 1000 which can be formed using only the digits 1, 1, 2, 3, 4, 0 taken four at a time is ______ .
- 16. [ Hint : 2 alike + zero + 1 different = 27 ; 2 alike + 2 non zero + different = 36 ; All four different '0' always included = 72 ; All four different non zero digit = 24 159 ] 163. The number of numbers that can be formed by using digits 3, 4, 5, 6, 5, 4, 3 so that the odd digits always occupy the odd places is : (A) 7 2 3 ! ( !) (B) 4 ! . 3 ! (C*) 4 3 2 3 ! . ! ( !) (D) none 164. Sum of all four digit numbers formed using only the digits 2, 2, 3, 1 is - [3] (A) 26644 (B*) 26664 (C) 39996 (D) 53328 165. Total number of divisors of 75600 which are divisible by 3 is ______ . [3 marks] [ Ans. : 90 ] 166. There are 10 seats in the first row of a theatre of which 4 are to be occupied . The number of ways of arranging 4 persons so that no two persons sit side by side is : [3 marks] (A) 7 C4 (B*) 4 . 7 P3 (C*) 7 C3 . 4 ! (D*) 840 [ Hint : 4 to be occupied say s s s s Remaining 6 are x x x x x x . Now 4 can be selected in 7 C4 ways and can be arranged in 7 C4 . 4 ! ways ] 167. The number of ways in which the number 94864 can be resolved as a product of two factors is : (A) 30 (B*) 23 (C) 45 (D) 46 [ Hint : 94864 = 24 . 72 . 112 ] 168.. A telegraph has 'm' arms and each arm is capable of 'n' distinct positions including the position of rest . The total number of signals that can be made is : [3] (A) m Pn (B) n Pm (C) mn - 1 (D*) nm - 1 (B) Selection 169. A box contains 6 balls which may be all of different colours or three each of two colours or two each of three different colours . The number of ways of selecting 3 balls from the box is : (A) 60 (B*) 31 (C) 30 (D) none [ Hint : Case I 6 C3 ; Case II A A A B B B (all three alike 2 ways or 2 alike and 1 different 2 ways = 4 ; Case III A A B B C C all three different 1 ; 2 alike + 1 different = 3 . 2 = 6 Total ways = 20 + 4 + 7 = 31) ] 170. 5 Indian & 5 American couples meet at a party & shake hands . If no wife shakes hands with her husband & no Indian wife shakes hands with a male , then the number of hand shakes that takes place in the party is : (A) 95 (B) 110 (C*) 135 (D) 150 [ Ans. 20 C2 (50 + 5) = 135 ] 171. Find the number of ways in which 3 numbers in A.P. can be selected from 1, 2, 3, ......n. [ Ans : n1 4 2 if n is odd , n n 2 4 if n is even ] 172. For a game in which two partners oppose two other partners, 6 men are available . If every possible pair must play every other pair, find the number of games played . [ Ans : 45 6 C4 . 3 ] 173. A man is dealt a poker hand comprising of five cards from an ordinary pack of 52 playing cards . Number of ways in which he can be dealt with a pair of aces and other three cards of different denominations, is : (A) 103776 (B*) 84480 (C) 84840 (D) 48840
- 17. [ Hint : 4 C2 . 12 C3 . 43 ] 174. 'A' is a set containing n elements . A subset P of 'A' is chosen . The set 'A' is reconstructed by replacing the elements of P . A subset 'Q' of 'A' is again chosen . The number of ways of choosing P and Q so that P Q contains exactly two elements is : (A) n C3 . 2n (B*) n C2 . 3n 2 (C) 3n 2 (D) none [ Hint : The two common elements can be selected in n C2 ways . Remaining (n 2) elements, each can be chosen in three way i.e. a P or a Q or a is neither in P nor in Q n C2 . 3n 2 ] 175. Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle . Then the maximum number of circles that can be drawn so that each contains atleast three of the given points is : (A) 216 (B*) 156 (C) 172 (D) none 5 (one circle) [ Hint : 11 6 (no 4 lies on the same circle) 5 C2 . 6 C1 + 6 C2 5 C1 + 6 C3 + 1 = 156 alternatively 12 C3 5 C3 + 1 ] 175. Six married couple are sitting in a room. Number of ways in which 4 people can be selected so that there is exactly one married couple among the four is : (A*) 240 (B) 255 (C) 360 (D) 480 [ Hint : Total = 12 C4 (exactly two married + All four different) = 12 C4 (6 C2 + 6 C4 . 16) = 240 . Alternatively : 6 C1 . 10 8 1 2 . . = 6 . 40 = 240 ] 176. The number of combination of 16 things, 8 of which are alike and the restdifferent, taken 8 at a time is _______. [ Ans. : 256 ] 177. The number of all possible selections of one or more questions from 10 given questions, each equestion having exactly two alternatives is : (A) 310 (B) 210 1 (C*) 310 1 (D) 210 [ Hint : 1st question can be selected in three ways and so on ] 178. How many committee of 3 boys & 4 Girls out of 8 boys & 7 Girls can be formed. If Boy x refuses to work with Girl y. (A) 1560 (B) 1960 (C*) 1540 (D) 1520 179. A candidate is required to answer 6 out of 10 questions which are divided into two groups each containing 5 questions, and he is not permitted to attempt more than 4 from each group. The number of different ways, he can make up his choice is ______ . [ Ans. 5 C4 . 5 C2 + 5 C3 . 5 C3 + 5 C2 . 5 C4 = 200 ] 180. In a school there are 11 students eligible to be appointed as monitors . Everyday a set of five members is to be appointed . The number of weeks that must elapse before the same set of five students will be in office as monitors is : (A) 462 (B) 77 (C) 62 (D*) 66 181. A jury of 10 is chosen from 7 men & 8 women . The number of juries on which women are in the majority is equal to : (A) 5 11 of the total (B) 6 13 of the total (C*) 61 143 of the total (D) none of these 182. If 28 C2r : 24 C2r 4 = 225 : 11 , then : (A) r = 24 (B) r = 14 (C*) r = 7 (D) none
- 18. 183. At an election, a voter may vote for any number of candidates not greater than the number to be chosen. There are 10 candidates and 5 members are to be chosen. The number of ways in which a voter may vote for at least one candidate, is given by : (A*) 637 (B) 638 (C) 639 (D) 640 184. Number of positive integral solutions of the equation 20 x + y + z 50 is: (A) 50 C3 – 20 C3 (B) 51 C3 – 19 C3 (C*) 50 C3 – 19 C3 (D) 51 C3 – 21 C3 185. The number of triplets (a, b, c) such that a b < c, where a, b, c {1, 2, 3,......... n} is (A*) n C3 + n C2 (B*) n+1 C3 (C) n–1 C3 (D) n–1 C2 + n–1 C3 186. The selection of 10 balls from an unlimited number of red, white, blue and green balls can be done in ' k ' ways where ' k ' is: (A*) 286 (B) 628 (C) 826 (D) 268 187. The number of ways in which a mixed double tennis game can be arranged from amongst 9 married couple if no husband & wife plays in the same game is: (A) 756 (B) 3024 (C*) 1512 (D) 6048 [ Hint: 9 C2 . 7 C2 . 2 ! = 1512 ] 188. John has x children by his first wife. Mary has (x + 1) children by her first husband. They marry and have children of their own. The whole family has 24 children. Assuming that two children of the same parents do not fight, find the maximum possible number of fights that can take place. Ans. 191 189. A woman has 11 close friends. Number of ways in which she can invite 5 of them to dinner, if two particular of them are not on speaking terms & will not attend together, is (A*) 11 C5 – 9 C3 (B*) 9 C5 + 2. 9 C4 (C*) 3.9 C4 (D) none of these 190. If 2 1 ) 1 r , n ( C ) r , n ( C and 3 2 ) 2 r , n ( C ) 1 r , n ( C then (A) n = 15 and r = 6 (B) n = 12 and r = 4 (C*) n = 14 and r = 4 (D) n = 14 and r = 6 191. A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if at least five women have to be included in committee. In how many of these committees (i) the women are in majority (ii) the men are in majority? 192. In a class there are 30 boys and 18 girls. The teacher wants to select one boy and one girl to represent the class for a quiz competition, the number of ways in which the teacher make this selection, are (A) 30! × 18! (B) 48! (C*) 540 (D) none of these 193. A rack has 5 different pairs of shoes. The number of ways in which 4 shoes can be chosen from it so that there will be no complete pair is: (A) 1920 (B) 200 (C) 110 (D*) 80 194. Number of ways in which we can choose 2 distinct integers from 1 to 100 such that the difference between them is atmost 10 is _______. [Ans. 100 C2 90 C2 ] 195. The number of all possible selections of one or more questions from 10 given questions, each equestion having exactly two alternatives is: (A) 310 (B) 210 1 (C*) 310 1 (D) 210 [ Hint: 1st question can be selected in three ways and so on ]
- 19. 196. There are books of x different subjects, each having y copies. Number of different selections, if atleast one book of each subject is to be included in each selection is ______. [ Ans. yx ] 197. A lady gives a dinner party for 5 guests. The number of ways in which they may be selected from among 9 friends if two of the friends will not attend the party together is: (A) 84 (B*) 91 (C) 133 (D) none 198. The number of combination of 16 things, 8 of which are alike and the restdifferent, taken 8 at a time is _______. [ Ans.: 256 ] 199. The number of ways in which we can choose 6 chocolates out of 8 different brands available in the market is:. (A*) 13 C6 (B) 13 C8 (C) 86 (D) none [ Hint: consider 8 different brands to be beggar and compute the distribution of 6 identical things among 8 people each receiving none, one or more. Alternatively find co-efficient of x6 in (1 + x + x2 +..... )8 ] 200. A lift with 7 people stops at 10 floors. People varying from zero to seven go out at each floor. The number of ways in which the lift can get emptied, assuming each way only differs by the number of people leaving at each floor, is: (A) 16 C6 (B) 17 C7 (C*) 16 C7 (D) none [ Hint: consider each floor to be a beggar. Now distribute 7 identical coins ( 7 people) in 10 beggars each receiving none, one or more 16 C7 ] 201. There are p intermediate railway stations on a route from one terminus to other. Number of ways in which a train can be stopped at 3 stations if no two stations are consecutive is _______. [ Ans.: p 2 C3 ] 202. You are given three classes of letters: (a1 , a2 , a3 , a4 ); (b1 , b2 , b3 ) and (c1 , c2 ). The total number of combinations which can be made with these letters when no two of the same class enter into any combination is: (A) 25 . 23 . 22 1 (B) 24 (C*) 59 (D) none 203. A box contains 6 balls which may be all of different colours or three each of two colours or two each of three different colours. The number of ways of selecting 3 balls from the box is: (A) 60 (B*) 31 (C) 30 (D) none [ Hint: Case I 6 C3 ; Case II A A A B B B (all three alike 2 ways or 2 alike and 1 different 2 ways = 4; Case III A A B B C C all three different 1; 2 alike + 1 different = 3. 2 = 6 Total ways = 20 + 4 + 7 = 31) ] 204. A question paper on mathematics consists of twelve questions divided into three parts A, B and C, each containing four questions. In how many ways can an examinee answer five questions, selecting atleast one from each part. (A) 623 (B*) 624 (C) 625 (D) 626 [ Hint: 3 (4 C2 . 4 C2 . 4 C1 ) + 3 (4 C1 . 4 C1 . 4 C3 ) = 432 + 194 = 624 ] 205. A woman has 11 colleagues in her office, of whom 8 are men. She would like to have some of her colleagues to dinner. Find the number of her choices, if she decides to invite (i) atleast 9 of them and (ii) all her women colleagues and sufficient men colleagues to make the number of women and men equal. [ Ans.: (a) (i) 11 c9 + 11 c10 + 11 c11 = 67 (ii) she has to invite 4 men, since there will be 4 women dinning, including herself. Hence the answer is 8 c4 ]
- 20. 206. A point in the cartesian plane whose coordinates are integers is called a lattice point. Consider a path from the origin to the lattice point A (m, n) where m & n are nonnegative, that, (i) starts from the origin (ii) is always parallel to the xaxis or the yaxis (iii) makes turns only at a lattice point, either along the positive xaxis or along the positive yaxis & (iv) terminates at A. Determine the number of such paths. [ Solution: A typical path is a sequence of 'm + n' unit steps, m of them horizontal and n of them vertical. Hence the number of paths is C (m + n, n) = C (m + n, n), the number of ways of reserving positions in the sequence for one or the other kind of step ] 207. A man is at the origin on the xaxis and takes a unit step either to the left or to the right. He stops after 5 steps or if he reaches 3 or 2. Number of ways in which he (A*) reaches 2 is 3 (B*) reaches 3 is 4 (C) stop exactly after walking 5 steps is 12 (D*) can perform the experiment is 20 [ for C the correct is 16 ] 208. In a certain algebraical exercise book there are 4 examples on arithmetical progressions, 5 examples on permutation combination and 6 examples on binomial theorem. Number of ways a teacher can select for his pupils atleast one but not more than 2 examples from each of these sets, is ______. [ Ans.: 3150 ] 209. The total number of combinations 6 at a time which can be formed from 6 alike white, 6 alike blue, 6 alike green & 6 alike red balls is: (A) 90 (B*) 84 (C) 78 (D) none [ Hint: [ 9 cases ] 0 0 0 0 0 0 0 0 0 = 9 C3 = 84 or co-eff. of x6 in (1 + x + x2 + x3 + x4 + x5 + x6 )4 Treat W, B, G, R as beggar] 210. The kindergarten teacher has 25 kids in her class. She takes 5 of them at a time, to zoological garden as often as she can, without taking the same 5 kids more than once. Then the number of visits, the teacher makes to the garden exceeds that of a kid by: (A*) 25 C5 24 C4 (B*) 24 C5 (C) 25 C5 24 C5 (D) 24 C4 [ You may use the fact that n Cr + n Cr 1 = n + 1 Cr ] 211. A jury of 10 is chosen from 7 men & 8 women. The number of juries on which women are in the majority is equal to: (A) 5 11 of the total (B) 6 13 of the total (C*) 61 143 of the total (D) none of these 212. 3 different railway passes are allotted to 5 students. The Number of ways in which it can be done is: (A*) 60 (B) 20 (C) 15 (D) 10 213. Six married couple are sitting in a room . Number of ways in which 4 people can be selected so that there is exactly one married couple among the four is : (A) 240 (B) 255 (C) 360 (D) 480 [ Hint : Total = 12 C4 (exactly two married + All four different) = 12 C4 (6 C2 + 6 C4 . 16) = 240 . Alternatively : 6 C1 . 10 8 1 2 . . = 6 . 40 = 240 ] 214. Find the number of 7 lettered words each consisting of 3 vowels and 4 consonants which can be formed using the letters of the word "DIFFERENTIATION". [6, 0] Ans. 532770 215. The number of ways selecting 8 books from a library which has 10 books each of Mathematics, Physics,
- 21. Chemistry and English, if books of the same subject are alike, is: (A) 13 C4 (B) 13 C3 (C) 11 C4 (D*) 11 C3 216. The result of 21 football matches (win, lose or draw) are to be predicted. The number of different forecasts which contain exactly 18 correct results is: (A) 21 C3 (B) 3. 21 C3 (C*) 8. 21 C3 (D) 27. 21 C3 217. A set contains (2n + 1) different elements. The number of subsets of the set which contains more than n elements is: (A) 2n 1 (B) 2n (C) 2n + 1 (D*) 4n 218. Let there be 9 fixed points on the circumference of a circle. Each of these points is joined to every one of the remaining 8 points by a straight line and the points are so positioned on the circumference that atmost 2 straight lines meet in any interior point of the circle. The number of such interior intersection points is: (A*) 126 (B) 351 (C) 756 (D) none of these 219. Rajdhani express going from Bombay to Delhi stops at 5 intermediate stations. 10 passengers enter the train during the journey with ten different ticket of two classes. The number of different sets of tickets they may have had is: (A) 15 C10 (B) 20 C10 (C*) 30 C10 (D) none 220. From the sequence of the first 20 natural numbers, four are selected such that they are not all consecutive. The number of such selections is: (A*) 284 17 (B) 285 17 (C) 284 16 (D) 285 16 221. A student has to answer 10 out of 13 questions in an examination. The number of ways in which he can answer if he must answer atleast 3 of the first five questions is: (A*) 276 (B) 267 (C) 80 (D) none [ Hint: 13 C10 number of ways in which he can reject 3 questions from the first five or 13 C10 5 C3 = 276 or 5 C3 . 8 C7 + 5 C4 . 8 C6 + 5 C5 . 8 C5 = 276 ] 222. Number of ways in which n students can be partitioned into two teams containing atleast one student is: (A) 2n 2 (B) 2n 2 (C*) 2n 1 1 (D) None 223. Number of ways in which a selection of 100 balls, can be made out of 100 identical red balls, 100 identical blue balls & 100 identical white balls can be made is: (A) 4950 (B) 5050 (C*) 5151 (D) none [ Ans. 102 C2 = 5151 use beggar 100 apples 3 beggars ] 224. The number of all possible selections of one or more questions from 10 given questions, each equestion having exactly two alternatives is: (A) 310 (B) 210 1 (C*) 310 1 (D) 210 [ Hint: 1st question can be selected in three ways and so on ] 225. The number of ways selecting 8 books from a library which has 10 books each of Mathematics , Physics , Chemistry & English , if books of same subject are alike , is : (A) 13 C4 (B) 13 C3 (C) 11 C4 (D*) 11 C3 [ Hint : Consider P, C, M and E to be four baggars and distribute 8 identical coins between them i.e. 11 C3 = 165 ] 226. A candidate is required to answer 7 questions out of 12 questions which are divided into 2 groups each containing 6 questions . He is not permitted to answer more than 5 from each group . In how many different ways , can he choose the 7 questions ? [ Ans. : 780 ] 227. A student has to answer 10 out of 13 questions in an examination . The number of ways in which he
- 22. can answer if he must answer atleast 3 of the first five questions is : (A*) 276 (B) 267 (C) 80 (D) none [ Hint : 13 C10 number of ways in which he can reject 3 questions from the first five or 13 C10 5 C3 = 276 or 5 C3 . 8 C7 + 5 C4 . 8 C6 + 5 C5 . 8 C5 = 276 ] (C) Cyclic Permutation 228. Number of ways in which 12 identical coins can be distributed in 6 different purses, if not more than 3 & not less than 1 coin goes in each purse is ______ . [ Hint : 000000 remaining (i) 2 coins in each of 3 purses = 6 C3 (selecting 3 purses from 6 different purses = 20 . (ii) 2 coins in one + 1 coin in 4 purses = 6 C1 . 5 C4 = 20 (iii) 2 coins in each of two purses + 1 coin in each of two purses = 6 C2 . 4 C2 = 90 (iv) 1 coin in each of 6 purses = 6 C6 = 1 or co-efficient of x12 in (x + x2 + x3 )6 = 141 ] 229. Number of ways in which 2 Indians , 3 Americans , 3 Italians and 4 Frenchmen can be seated on a circle , if the people of the same nationality sit together is : (A) 2 . (4 !)2 (3 !)2 (B*) 2 . (3 !)3 . 4 ! (C) 2 . (3 !) (4 !)3 (D) none 230. The number of ways in which 10 boys can take positions about a round table if two particular boys must not be seated side by side is : (A) 10 (9) ! (B) 9 (8) ! (C*) 7 (8) ! (D) none 231. The number of ways in which 10 identical apples can be distributed among 6 children so that each child receives atleast one apple is : (A*) 126 (B) 252 (C) 378 (D) none of these 232. There are 5 balls of different colours & 5 boxes of colours same as those of the balls . The number of ways in which the balls, one in each box, could be placed such that exactly 2 balls go to the box of its own colour is : (A) 31 (B) 30 (C*) 20 (D) none 233. The number of ways in which 10 apples , 5 oranges & 5 mangoes can be distributed among three persons, each receiving none, one or more is : (A*) 12 C2 . 7 C2 . 7 C2 (B) 13 C3 . 8 C3 . 7 C3 (C) ( ) ! ! ! ! ! 22 10 5 5 2 (D) none 234. Let m denote the number of ways in which 4 different books are distributed among 10 persons, each receiving none or one only and let n denote the number of ways of distribution if the books are all alike . Then : (A) m = 4n (B) n = 4m (C*) m = 24n (D) none [ Hint : m = 10 C4 . 4 ! and n = 10 C4 ] 235. Number of ways in which two Americans, two British , one Chinese , one Dutch and one Egyptian can sit on a round table so that persons of the same nationality are separated is : (A) 48 (B) 240 (C*) 336 (D) none [ Hint : Total = 6 ! n (A) = when A1 A2 together = 5 ! 2! n (B) = when B1 B2 together = 5 ! 2 ! So n (A Ç B) = 4 ! 2 ! 2 ! = 96 Hence n A B = 6 ! [ 2 . 5 ! 2 ! - 96 ] = 720 - 384 = 336 ] [DPP-45(A)/Batch-P/Q.No.10] 236. Number of ways in which 4 girls & 4 boys can sit around in circular table if the boys & girls are alternate & a particular boy & a girl are never together in any arrangement is : (A) 108 (B) 96 (C) 84 (D*) 72
- 23. [DPP-43(A)/Batch-P/Q.No.5] 237. In how many different ways may 12 things 4 each of three varieties be distributed equally among two persons? Things of the same variety are assumed to be identical. [Ans. 19] 238. Number of ways in which 4 girls & 4 boys can sit around in circular table if the boys & girls are alternate & a particular boy & a girl are never together in any arrangement is: (A) 108 (B) 96 (C) 84 (D*) 72 239. Number of different ways in which 8 different books can be distributed among 3 students, if each student receives at least 2 books is......................... Ans. 2940 240. Seven different coins are to be divided amongst three persons. If no two of the persons receive the same number of coins but each receives at least one coin & none is left over, then the number of ways in which the division can be made is: (A) 420 (B*) 630 (C) 710 (D) none of these 241. The number of ways in which 5 beads, chosen from 8 different beads be threaded on to a ring is: (A*) 672 (B) 1344 (C) 336 (D) none of these 242. There are 12 different marbles to be divided between two children in the ratio 1: 2. The number of ways it can be done is: (A*) 990 (B) 495 (C) 600 (D) none 243. Number of ways in which 9 different prizes be give to 5 students if one particular boy receives 4 prizes and the rest of the students can get any numbers of prizes, is: (A*) 9C4. 210 (B) 9C5. 54 (C) 4. 45 (D) none 244. The number of ways in which 5 different prizes be given among 11 candidates, each candidate receiving any number of prizes, are (A*) 115 (B) 511 (C) 11C5 (D) 11P5 245. There are 5 balls of different colours & 5 boxes of colours same as those of the balls. The number of ways in which the balls, one in each box, could be placed such that exactly 2 balls go to the box of its own colour is: (A) 31 (B) 30 (C*) 20 (D) none 246. Let m denote the number of ways in which 4 different books are distributed among 10 persons, each receiving none or one only and let n denote the number of ways of distribution if the books are all alike. Then: (A) m = 4n (B) n = 4m (C*) m = 24n (D) none [ Hint: m = 10 C4 . 4 ! and n = 10 C4 ] 247. Number of ways in which p identical white balls, q identical black balls & r identical red balls can be put in n different bags, if one or more of the bags remain empty is ______. [ Ans. n + p 1 Cp . n + q 1 Cq . n + r 1 Cr beggar ] 248. Number of ways in which all the letters of the word "ALASKA " can be arranged in a circle distinguishing between the clockwise and anticlockwise arrangement, is: (A) 60 (B) 40 (C*) 20 (D) none of these [ Hint: Consider A 's to be different Number of ways (6 1) ! = 120 the A1 , A2 & A3 can be arranged in 3 ! ways
- 24. 120 6 = 20 ] 249. The number of ways in which 10 boys can take positions about a round table if two particular boys must not be seated side by side is: (A) 10(9)! (B) 9(8)! (C*) 7(8)! (D) none 250. There are n persons among whom 2 are brothers. The number of ways they can be arranged in a circle, if there is exactly one person between the two brothers is ______. [ Ans. ( n 3) ! (n 2). 2 ! = 2. (n 2) ! ] 251. 6 white and 6 black balls are distributed among ten identical urns so that there is atleast one ball in each urn. Balls are all alike except for the colour and each box can hold any number of balls. The number of different distributions of the balls is: (A) 26250 (B) 132 (C) 12 (D*) 10 [ Hint: (i) 3 balls in one urn + 1 ball in each of 3 urns (ii) 2 balls in each of two urns + 1 ball in each of 8 urns (i) B B B ; W W W; W B B W W B = 4 (ii) B B B B; B B B W; B B W W; W W W W; W W W B; W B W B = 6 10 ] 252. Number of ways in which two Americans, two British, one Chinese, one Dutch and one Egyptian can sit on a round table so that persons of the same nationality are separated is: (A) 48 (B) 240 (C*) 336 (D) none [ Hint: Total = 6 ! n (A) = when A1 A2 together = 5 ! 2! n (B) = when B1 B2 together = 5 ! 2 ! So n (A B) = 4 ! 2 ! 2 ! = 96 Hence n A B = 6 ! [ 2. 5 ! 2 ! 96 ] = 720 384 = 336 ] 253. The number of ways in which 21 apples can be shared among 4 persons, each receiving not less than 3 is: (A) 180 (B*) 220 (C) 260 (D) none 254. There are (p + q) different books on different topics in Mathematics. (p q) If L = The number of ways in which these books are distributed between two students X and Y such that X get p books and Y gets q books. M = The number of ways in which these books are distributed between two students X and Y such that one of them gets p books and another gets q books. N = The number of ways in which these books are divided into two groups of p books and q books then, (A) L = M = N (B) L = 2M = 2N (C*) 2L = M = 2N (D) L = M = 2N 255. Number of ways in which two Americans, two British , one Chinese , one Dutch and one Egyptian can sit on a round table so that persons of the same nationality are separated is : [3 – 1] (A) 48 (B) 240 (C*) 336 (D) none [ Hint : Total = 6 ! n (A) = when A1 A2 together = 5 ! 2! n (B) = when B1 B2 together = 5 ! 2 ! So n (A Ç B) = 4 ! 2 ! 2 ! = 96 Hence n A B = 6 ! [ 2 . 5 ! 2 ! - 96 ] = 720 - 384 = 336 ]
- 25. 256. Number of ways in which 9 different prizes be give to 5 students if one particular boy receives 4 prizes and the rest of the students can get any numbers of prizes, is : [3, –1] (A*) 9C4 . 210 (B) 9C5 . 54 (C) 4 . 45 (D) none 257. 12 guests at a dinner party are to be seated along a circular table. Supposing that the master and mistress of the house have fixed seats opposite one another, and that there are two specified guests who must always, be placed next to one another; the number of ways in which the company can be placed, is: (A*) 20. 10 ! (B) 22. 10 ! (C) 44. 10 ! (D) none 258. Number of different ways in which 8 different books can be distributed among 3 students, if each student receives atleast 2 books is ______. [Hint: 8 books can be distributed in a group of (2, 2, 4) or (2, 3, 3). Number of groups are 8 2 2 4 2 8 2 3 3 2 ! ! ! ! ! ! ! ! ! ! & can be distributed in 3 ! ways; Ans. 2940 ] 259. The number of ways of distributing 12 identical oranges among 4 children so that every child gets at least one and no child more than 4 is ______. (co-efficient of x12 in (x + x2 + x3 + x4 )4 [ Ans. 31 ] 260. If 'D' denotes the number of ways in which 8 different things be distributed between Ram and Shyam and 'L' denotes the corresponding figure when the things are all alike, each person receiving atleast one thing in both the cases then, D L = ______. [ Ans.: 254 7 = 247; Hint: (28 2) 7 ] 261. The number of ways of distributing 12 identical oranges among 4 children so that every child gets at least one and no child more than 4 is ______. [ Ans. 31 ] (co-efficient of x12 in (x + x2 + x3 + x4 )4 262. Number of ways in which 10 children sit in a merrry go round relatively to one another is ______. [ Ans.: 9 ! ] (D) Grouping 263. The number of dissimilar terms in the expansion of (a + b + c)n is : [3 marks] (A) n n ( ) 1 2 (B*) ( ) ( ) n n 1 2 2 (C) ( ) ( ) n n 2 3 2 (D) none 264. Seven different coins are to be divided amongst three persons . If no two of the persons receive the same number of coins but each receives atleast one coin & none is left over , then the number of ways in which the division may be made is : (A) 420 (B*) 630 (C) 710 (D) none 265. The number of ways in which 200 different things can be divided into groups of 100 pairs is : (A) 200 2100 ! (B*) 101 2 102 2 103 2 .... 200 2 (C*) 200 2 100 100 ! ( ) ! (D*) (1 . 3 . 5 . ..... 199) 266. In an eleven storey building 9 people enter a lift cabin . It is known that they will leave the lift in groups of 2, 3 and 4 at different residential storeys . Number of ways in which they can get down is :
- 26. (A) 12 2 3 4 2 ! ! ! . ! (B*) 10 4 ! (C) 10 4 ! ! (D) none [ Hint : 9 people can be divided into groups of 2, 3 and 4 in 9 2 3 4 ! ! ! ! ways . Now each group can be distributed in 10 C3 . 3 ! ways 9 2 3 4 ! ! ! ! 10 . 9 . 8 = 10 4 ! Note that in an eleven storeyed building there will be 10 floors ] 267. Number of different ways in which 8 different books can be distributed among 3 students, if each student receives atleast 2 books is ______ . [ Hint : 8 books can be distributed in a group of (2 , 2 , 4) or (2 , 3 , 3) . Number of groups are 8 2 2 4 2 8 2 3 3 2 ! ! ! ! ! ! ! ! ! ! & can be distributed in 3 ! ways ; Ans. 2940 ] 268. The number of ways in which 10 apples, 5 oranges & 5 mangoes can be distributed among three persons, each receiving none, one or more is: (A*) 12 C2 . 7 C2 . 7 C2 (B) 13 C3 . 8 C3 . 7 C3 (C) ( ) ! ! ! ! ! 22 10 5 5 2 (D) none 269. In a certain college at the B.Sc. examination, 3 candidates obtained first class honours in each of the following subjects: Physics, Chemistry and Maths, no candidates obtaining honours in more than one subject; Number of ways in which 9 scholarships of different value be awarded to the 9 candidates if due regard is to be paid only to the places obtained by candidates in any one subject is __________. [ Ans: 9 3 3 ! ( !) = 1680 ] 270. Number of ways in which a pack of 52 playing cards be distributed equally among four players so that each may have the Ace, King, Queen and Jack of the same suit is: (A) 36 9 4 ! ! (B*) 36 4 9 4 ! . ! ! (C) 36 9 4 4 ! ! . ! (D) none [ Hint: divide 36 cards in 4 equals and distribute in 4 ! and A, K, Q, J of same suit also in 4 ! ways ] 271. A committee of 10 is to be formed from 8 teachers and 12 students of whom 4 are girls. In how many ways this can be done so that the committee contains atleast four of either groups (teachers and students) and atleast 2 girls and atleast 2 boys are in the committee. [ Solution: 8 Teacher 20 People Committee of 10 12 Students atleast atleast 4 G 8 B 4 teachers 4 students atleast atleast 2 G 2 B We can have (i) 4 teachers + 6 students 4 T and [ (2 G, 4 B) + (3 G + 3 B) + (4 G + 2 B) ] (ii) 5 teachers + 5 students 5 T and [ (2 G, 3 B) + (3 G + 2 B) ] (iii) 6 teachers + 4 students 6 T and (2 G, 2 B) Now (i) + (ii) + (iii) = 8 C4 [ 4 C2 . 8 C4 + 4 C3 . 8 C3 + 4 C4 . 8 C2 ] + 8 C5 [ 4 C2 . 8 C3 + 4 C3 . 8 C2 ] + 8 C2 [ 4 C2 . 8 C2 ] = 47040 + 25088 + 4704 = 76832 (Ans.) ] 272. A train time-table must be compiled for various days of the week so that two trains a day depart for
- 27. three days, one train a day for two days and three trains a day for two days. How many different time- tables can be compiled ? (A) 140 (B*) 210 (C) 133 (D) 72 Sol. The number of trains a day (the digits 1, 2, 3) are three groups of like elements from which a sample must be formed. In the time-table for a week the number 1 is repeated twice, the number 2 is repeated 3 times and the number 3 is repeated twice. The number of different time-tables is equal to: p (2, 3, 2) = 7 2 3 2 ! ! ! ! = 210 ] 273. Seven different coins are to be divided amongst three persons. If no two of the persons receive the same number of coins but each receives atleast one coin & none is left over, then the number of ways in which the division may be made is: (A) 420 (B*) 630 (C) 710 (D) none [ Hint: 7 1 2 4 ! ! ! ! x 3 ! ] 274. The number of ways in which 10 students be divided into three teams one containing 4 students & the other 3 is: (A) 10 4 3 3 ! ! ! ! (B*) 2100 (C) 10 C4 . 5 C3 (D) 10 6 3 3 ! ! ! ! 1 2! 275. A candidate is required to answer 6 out of 10 questions which are divided into two groups each containing 5 questions, and he is not permitted to attempt more than 4 from each group. The number of different ways, he can make up his choice is ______. [ Ans. 5 C4 . 5 C2 + 5 C3 . 5 C3 + 5 C2 . 5 C4 = 200 ] 276. The number of ways in which 200 different things can be divided into groups of 100 pairs is: (A) 200 2100 ! (B*) 101 2 102 2 103 2 .... 200 2 (C*) 200 2 100 100 ! ( ) ! (D*) (1. 3. 5...... 199) 277. How many committee of 3 boys & 4 Girls out of 8 boys & 7 Girls can be formed. If Boy x refuses to work with Girl y. (A) 1560 (B) 1960 (C*) 1540 (D) 1520 278. 9 persons enter a lift from ground floor of a building which stops in 10 floors (excluding ground floor). If is known that persons will leave the lift in groups of 2, 3, & 4 in different floors. In how many ways this can happen. [4, 0] Ans. 907200 279. Out of 16 players of a cricket team, 4 are bowlers and 2 are wicket keepers. A team of 11 players is to be chosen so as to contain at least 3 bowlers and at least 1 wicketkeeper. The number of ways in which the team be selected is (A) 2400 (B*) 2472 (C) 2500 (D) 960 280. How many committee of 3 boys & 4 Girls out of 8 boys & 7 Girls can be formed. If Boy x refuses to work with Girl y. (A) 1560 (B) 1960 (C*) 1540 (D) 1520 (E) Selection of one or more objects
- 28. 281. The number of ways in which 15 apples & 10 oranges can be distributed among three persons , each receiving none , one or more is : (A) 5670 (B) 7200 (C*) 8976 (D) none of these 282. The number of ways in which 21 apples can be shared among 4 persons, each receiving not less than 3 is : (A) 180 (B*) 220 (C) 260 (D) none (F) Multinomial theorem 283. The number of times the digit 3 will be written when listing the integers from 1 to 1000 is : (A*) 300 (B) 269 (C) 271 (D) 302 [ Hint : 0 xyz 9 (3 C1 . 92 ) 1 + (3 C2 . 9) 2 + 3 C3 . 3 = 300 ] 284. The number of integral solutions of the equation, x1 + x2 + x3 > 0 if xi 5, i = 1, 2, 3 is ______ . [ Ans. xi + 5 0 y1 + y2 + y3 = 15 17 C2 ] 285. The number of non negative integral solution of the equation, x + y + 3z = 33 is : (A) 120 (B) 135 (C*) 210 (D) 520 [ Hint : consider cases when z = 0, 1, 2, ....... , 11 34 + 31 + 28 + ...... + 1 (12 times) = 12 2 (1 + 34) = 210 ] 286. In an examination, the marks for physics and chemistry papers are 25 each, whereas maximum marks for maths paper is 100 . The number of ways in which a student can score 50 marks is : [assume that the marks are awarded in non-negative integral values] (A) 716 (B*) 676 (C) 605 (D) none of these [ Hint : coefficient of x50 in (1 + x + x2 + ...... x25 )2 (1 + x + x2 + ...... + x100 ) 287. The number of ways of distributing 12 identical oranges among 4 children so that every child gets at least one and no child more than 4 is ______ . (co-efficient of x12 in (x + x2 + x3 + x4 )4 [ Ans. 31 ] 288. The total number of combinations 6 at a time which can be formed from 6 alike white, 6 alike blue, 6 alike green & 6 alike red balls is : (A) 90 (B*) 84 (C) 78 (D) none [ Hint : [ 9 cases ] 0 0 0 0 0 0 0 0 0 = 9 C3 = 84 or co-eff. of x6 in (1 + x + x2 + x3 + x4 + x5 + x6 )4 Treat W, B, G, R as beggar] 289. The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question, is.............. [Ans. 21 C7 ] 290. Number of integral solutions of the equation 3 x + y + z = 27, where x, y, z > 0 is _____. [ Ans. 100 ] [ Hint: x = 1, 2,........ 8 and use beggar ] 291. The number of non negative integral solution of the equation, x + y + 3z = 33 is: (A) 120 (B) 135 (C*) 210 (D) 520 [ Hint: consider cases when z = 0, 1, 2,......., 11 34 + 31 + 28 +...... + 1 (12 times) = 12 2 (1 + 34) = 210 ] 292. In an examination, the marks for physics and chemistry papers are 25 each, whereas maximum marks
- 29. for maths paper is 100. The number of ways in which a student can score 50 marks is: [assume that the marks are awarded in non-negative integral values] (A) 716 (B*) 676 (C) 605 (D) none of these [ Hint: coefficient of x50 in (1 + x + x2 +...... x25 )2 (1 + x + x2 +...... + x100 ) 293. Six cards are drawn one by one from a set of unlimited number of cards, each card is marked with numbers 1, 0 or 1 . Number of different ways in which they can be drawn if the sum of the numbers shown by them vanishes, is : [3, – 1] (A) 111 (B) 121 (C*) 141 (D) none [ Hint : 0 0 0 0 0 0 1 way ; 0 0 0 0 1, 1 ! 4 ! 6 = 30 ways ; 0 0 1 1 1 1 ! 2 ! 2 ! 2 ! 6 = 90 ways ; 1 1 1 1 1 1 ! 3 ! 3 ! 6 = 20 ways Total = 141 Alternatively co-efficient of x0 in (1 + x 1 + x2 )6 ] 294. Find the number of non negative integral solution of the system of equations x1 + x2 + x3 + x4 + x5 = 20 and x1 + x2 + x3 = 5 Ans. 336 295. The number of integral solutions of the equation, x1 + x2 + x3 = 0 if xi 5, i = 1, 2, 3 is _________. [ Ans. xi + 5 0 y1 + y2 + y3 = 15 17 C2 ] 296. The number of non negative integral solution of the equation x+y+z+w 7 is ______ . [ Hint : find x + y + z + w + x = 7 = 11 C4 = 330 Ans. ] 297. Number of integral solutions of the equation 3 x + y + z = 27, where x , y , z > 0 is _____. [3 marks] [ Ans. 100 ] [ Hint : x = 1, 2, ........ 8 and use beggar ] (G) Dearrangement 298. The number of ways in which 200 things can be divided into groups of 100 pairs is : (A) 200 2100 ! (B*) 101 2 102 2 103 2 .... 200 2 (C*) 200 2 100 100 ! ( ) ! (D*) (1 . 3 . 5 . ..... 199) (H) Application to geometrical problem 299. In a plane there are 3 straight lines concurrent at a point 'P', 4 others which are concurrent at a point Q and 5 others which are concurrent at a third point R . Supposing no other three intersect at any point and no two are parallel then the number of triangles that can be formed by the intersection of these straight lines is : (A) 60 (B) 180 (C*) 205 (D) none [ Hint : 12 C3 - (3 C3 + 4 C3 + 5 C3 ) = 205 ] 300. The number of ways in which 5 X's can be placed in the squares of the figure so that no row remains empty is : (A) 56 (B*) 44 (C) 98 (D) 40
- 30. [ Hint : 8C5 [ only top remains exmpty + bottom empty] 8C5 – [6C5 + 6C5 ] = 56 – 12 = 44 [DPP-43(A)/Batch-P/Q.No.7] 301. In the given 4 × 3 square Grid. In how many ways can we choose squares of same dimension: (A) 96 (B*) 82 (C) 48 (D) 66 302. The number of ways in which 5 X's can be placed in the squares of the figure so that no row remains empty is: (A) 56 (B*) 44 (C) 98 (D) 40 [ Hint: 8C5 [ only top remains exmpty + bottom empty] 8C5 – [6C5 + 6C5 ] = 56 – 12 = 44 303. A polygon has 170 diagonals. How many sides it will have ? (A) 12 (B) 17 (C*) 20 (D) 25 304. Number of sub parts into which ' n ' straight lines in a plane can divide it is: (A*) n n 2 2 2 (B) n n 2 4 2 (C) n n 2 6 2 (D) none [Hint: When n = 1, 2, 3, 4,..... the corresponding number of parts in which the plane is divided are 2, 4, 7, 11...... Now find Tn of this series. 305. Number of ways in which A A A B B B can be placed in the squares of the figure as shown, so that no row remains empty, is: (A) 2430 (B) 2160 (C*) 1620 (D) none [ Hint: 2nd & 4th row block has to be selected. For remaining 4 letters they can be filled in 2, 1, 1 combination i.e. 3 ways. Hence total selections = 3. 3 C2 . 3 C1 . 3 C1 = 81 Number of ways of filling = 81 6 3 3 ! ! ! = 81 20 = 1620 ] 306. m points on one straight line are joined to n points on another straight line. The number of points of intersection of the line segments thus formed is [5, –1] (A*) m C2 . n C2 (B*) 4 ) 1 n )( 1 m ( mn (C) 2 C . C 2 n 2 m (D) m C2 + n C2 307. The number of ways in which 5 X's can be placed in the squares of the figure so that no row remains empty is:
- 31. (A) 97 (B*) 44 (C) 100 (D) 126 308. A regular polygon has 104 diagonals. The number of sides is __________ Ans : 16 309. There are 12 points in a plane of which 5 are collinear. The maximum number of distinct quadrilaterals which can be formed with vertices at these points, is: (A*) 2. 7 P3 (B) 7 P3 (C) 10. 7 C3 (D*) 420 310. In a plane, a set of 8 parallel lines intersect a set of ‘n’ parallel lines, that goes in another direction, forming a total 1260 parallelograms. The value of ‘n’ is: (A) 6 (B*) 10 (C) 8 (D) 12 311. Number of sub parts into which ' n ' straight lines in a plane can divide it is : (A*) n n 2 2 2 (B) n n 2 4 2 (C) n n 2 6 2 (D) none Hint : When n = 1, 2, 3, 4, ..... the corresponding number of parts in which the plane is divided are 2, 4, 7, 11 ...... Now find Tn of this series . 312. A polygon has 170 diagonals . How many sides it will have ? (A) 12 (B) 17 (C*) 20 (D) 25 313. Number of ways in which A A A B B B can be placed in the squares of the figure as shown , so that no row remains empty , is : (A) 2430 (B) 2160 (C*) 1620 (D) none [ Hint : 2nd & 4th row block has to be selected . For remaining 4 letters they can be filled in 2, 1, 1 combination i.e. 3 ways . Hence total selections = 3 . 3 C2 . 3 C1 . 3 C1 = 81 Number of ways of filling = 81 6 3 3 ! ! ! = 81 20 = 1620 ] (I) Rank 314. The letters of work ‘RANDOM’ are written in all possible orders in a dictionary, then rank of word RANDOM is : (A*) 614 (B) 641 (C) 461 (D) 613 315. If letters of the word "PARKAR" are written down in all possible manner as they are in a dictionary, then the rank of the word "PARKAR" is ______ . [ Ans. : 99 ] 316. A library has ' a ' copies of one title, ' b ' copies each of two titles, ' c ' copies each of three titles and single copy of ' d ' title . The number of ways in which the books can be arranged in a row is : (A) ( ) ! ! ! ! a b c d a b c (B) ( ) ! ! ( !) ( !) a b c d a b c 2 3 (C*) ( ) ! ! ( !) ( !) a b c d a b c 2 3 2 3 (D) none of these 317. The letters of the word TOUGH are written in all possible orders & these words are written out as in a dictionary, then the rank of the word TOUGH is : (A) 120 (B) 88 (C*) 89 (D) 90
- 32. 318. All the 7 digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once, and not divisible by 5 are arranged in the increasing order. Find the (2001)st number in this list. [ Ans.: 4315726 ] [ Solution: Total numbers which are not divisible by 5 are = 6 ! 6 = 4320 Now when 1 or 2 or 3 occupies the 7th place, then the number of numbers = 3 5 ! 5 = 1800 (last can be filled only in 5 ways) when 1st two places are 41.......... then 4 1 number of numbers = 4 ! 4 = 96 4 ways with 42.......... 4 2 number of numbers = 4 ! 4 = 96 4 ways Total so far = 1800 + 192 = 1992. 1st three places are filled as 4 3 1 2 4 3 1 2.......... 2 ways number of numbers = 2 ! 2 = 4 [ Total = 1992 + 4 = 1996 ] Now, when first 4 places are, 4 3 1 5 then the remaining 3 places in each case be filled in 3 ! = 6 ways which makes total numbers = 2002 and the (2002)th number is 4315762 Hence (2001)st number is just before it = 4315726 (Ans) ] 319. If letters of the word "PARKAR" are written down in all possible manner as they are in a dictionary, then the rank of the word "PARKAR" is: (A) 98 (B*) 99 (C) 100 (D) 101 320. The letters of word ‘RANDOM’ are written in all possible orders in a dictionary, then rank of word RANDOM is: (A*) 614 (B) 641 (C) 461 (D) 613 321. If all letters of word PAPAD are arranged in a dictionary, then the rank of work PAPAD is: (A) 19th (B*) 23rd (C) 22nd (D) none 322. If the letters of the word ‘SHWETA’ are written in all possible ways and then are arranged as in a dictionary, then the rank of the word ‘SHWETA’ is.................. [Ans. 430] (J) Misc. (K) Comprehension (L) Subjective 323. How many natural numbers are there from 1 to 1000 (including 1 & 1000) (i) Which have none of their digits repeated. (ii) Which have at least one digits repeated. Ans. (i) 738 (ii) 1000 – 738 324. There 6 pairs of different gloves. In how many different ways can each of 6 persons be distributed a pair of left and right gloves, if atleast 3 persons get a complete pair. [6, 0] Ans. 8 !
- 33. 325. From 25 tickets numbered from 1 to 25, number of ways in which 3 tickets can be chosen such that the numbers on them are in A.P. with even common difference, is ______ . [6, 0] [ Ans. : 66 ] [ Hint : if d = 2 21 ; if d = 4 17 ; d = 6 13 etc. Hence 1 + 5 + 9 + ...... + 17 + 21 = 66 ] 326. In how many other ways can the letters of the word MULTIPLE be arranged ; (i) without changing the order of the vowels (ii) keeping the position of each vowel fixed (iii) without changing the relative order/ position of vowels & consonants. [6, 0] Ans. (i) 3359 (ii) 59 (iii) 359 327. In how many ways an insect can move from left bottom corner of a chess board to the right top corner, if it is given that it can move only upside or right, along the lines. [4, 0] Ans. 16 C8 328. A man has 3 friends. In how many ways he can invite one friend everyday for dinner on 6 successive nights so that no friend is invited more than 3 times. [6, 0] 329. The number of 5 digit numbers of the form x y z y x in which x < y is : (A) 350 (B*) 360 (C) 380 (D) 390 [ Hint : The first digit ' x ' can be any one of from 1 to 8 where as ' z ' can be any one from 0 to 9 when x = 1 , y = 2, 3, 4, ...... , 9 x = 2 , y = 3, 4, ...... , 9 and so on Thus the total = (8 + 7 + 6 + ...... + 2 + 1) 10 = 360 ] 330. The sum of all the four digit even numbers which can be formed by using the digits 0, 1, 2, 3, 4 and 5 if repetition of digits is allowed is: (A) 1765980 (B) 1756980 (C*) 1769580 (D) 1759680 331. The Reserve Bank of India prints currency notes in denominations of Five rupees, Ten rupees, Twenty rupees, Fifty rupees, One hundred rupees, Five hundred rupees and One thousand rupees. In how many ways can it display in a set order, ten currency notes not necessarily of different denominations ? How many of these will have currency notes of all denominations ? [ Ans.: 710 ; 49 6 10 ! ] A: 5 rupees B: 10 rupees 1st place we have 7 choices C: 20 rupees 2nd place we have 7 choices etc D: 50 rupees E: 100 rupees Hence total number of ways F: 500 rupees 10 places can be filled = 710 Ans. G: 1000 rupees For currency notes of all denomination A B C D E F G exactly one currency repeats eg. A B C D E F G A A A exactly two currency repeats eg. A B C D E F G A B B & A B C D E F G A A B exactly three currency repeats eg. A B C D E F G A B C Now (i) = 7 C1 10 4 ! ! = 7 10 4 . ! !
- 34. (ii) = 7 C2 . 2 10 2 3 ! ! ! = 7 10 2 . ! ! (iii) = 7 C3 . 10 2 2 2 ! ! ! ! = 35 10 8 . ! Total = 10 ! 7 24 7 2 35 8 = 10 ! 7 84 105 24 = 196 24 10 ! = 49 6 10 ! (Ans.) 332. How many 5 digit numbers are there which contains not more than two different digits ? [ Ans.: 1224 ] Sol. excluding the digit ' 0 ', two digits out of the remaining 9 can be selected in 9 C2 ways. e.g. 1, 2, 3, 4 etc. Now all the 5 digit numbers which can be made which do not contain all digits identical = 9 C2 (25 2) ( like 1 1 1 1 1 ) But we have 9 such numbers containing all alike digits. Hence total 5 digit numbers nore of them containing the digit ' 0 ' having not more than two alike digits, = 9 C2 (25 2) + 9 = 1080 + 9 = 1089 Now with ' 0 ' always included, we have: 0 1 1 1 1 etc = 9 5 4 1 ! ! = 36 0 0 1 1 1 etc = 9 5 3 2 4 3 ! ! ! ! ! = 54 0 0 0 1 1 etc = 9 5 3 2 4 2 3 ! ! ! ! ! ! = 36 0 0 0 0 1 etc = 9 = 9 = 135 Total = 1089 + 135 = 1224 ] Aliter Select the remaining 1 digit in 9 C1 ways e.g. 01, 02 etc 9 C1 (1 24 ) 9 (2) (1) + (2) 1080 + 144 = 1224 (Ans.) ] 333. There are 2n guests at a dinner party. Supposing that the master an d mistress of the house have fixed seats opposite one another, and that there are two specified guests who must not be placed next to one another. Show that the number of ways in which the company can be placed is (2n 2) !. (4n2 6n + 4). [ Hint: Excluding the two specified guests, 2 n persons can be divided into two groups one containing n and the other (n 2) in ( ) ! ! ( )! 2 2 2 n n n and can sit on either side of Master & Mistress in 2 ! ways and can arrange themselves in n! (n 2)! Now the two specified guests where (n 2) guests are seated will have (n 1) gaps and can arrange themselves in 2 ! ways. Number of ways when G1 G2 will always be together
- 35. = ( ) ! ! ( )! 2 2 2 n n n 2 ! n ! (n 2) ! (n 1). 2! = (2 n 2) !. 4 (n 1) Hence number of ways when G1 G2 are never together = 2 2 ! ! ! ! n n 2 !. n !. n ! 4 (n 1). (2 n 2) ! = (2 n 2) ! 2 2 1 4 1 n n n ( ) ( ) = (2 n 2) ! 4 6 4 2 n n ] 334. Number of ways in which we can choose 2 distinct integers from 1 to 100 such that the difference between them is atmost 10 is _______. [Ans. 945] 335. There are books of x different subjects, each having y copies . Number of different selections, if atleast one book of each subject is to be included in each selection is ______. [ Ans. yx ]