Math Statistic study about permutation and combination.

1. Math Statistic
Cambridge University Press
By: danica p
Permutation
and
Combination

2. Permutation, what s it?
Permutation is an arrangement of elements of a set in which the order of the
elements considered.
Means, it explain the number of ways in which a subset of objects can be
selected from a given set of objects, where order is important.
Example:
Given the set of three letters, {A, B, C}, how many possibilities are there for selecting
any two letters where order is important?
Answer: (AB, AC, BC, BA, CA, CB) there is 6 arrangements are possible.
from A:
A-B, A-C (2 arrangements)
From B:
B-A, B-C (2 arrangements)
From C:
C-A, C-B (2 arrangements) Total: 2*3 = 6 arrangements
Notice!
PERMUTATION (P)
=
ARRANGEMENT !

3. Factorial Formula of Permutation
!
.
( )!
n r
n
P
n r Notation of n = n!
n! = n*(n-1)*(n-2)*.......*2*1
e.g. 9!=9*8*7*6*5*4*3*2*1
* = multiply
Example:
15 runners are hoping to take part in a marathon competition, but the track has only 4
lines. How many ways can 4 of the 15 runers be assigned to lanes?
Answer: Using Factorial formula of P,
15P4 = = 32760
1*2*3*4
1*2*3*4*5*6*7*8*9*10*11*12*13*14*15
)!1115(
!15
You can using your scientific
calcuator, it simple, you enter number n,
then shift-x-then enter the number of r, try it!
15-shift-x-P- then, 4, so it became to 15P4 =
32760

4. Then, what about a combination?
Combination is an arrangement of elements of a set in which the order of
elements is not considered.
Means: The number of ways in which a subset of objects can be selected from a
given set of objects, where order is not important.
Example:
Given the set of three letters, {A, B, C}, how many possibilities are there for
selecting any two letters where order is not important?
Answer: (AB, AC, BC) 3 combinations.
each alphabet has different couple combination, and that is:
A can together with B,
A can together with C,
B can together with C, total: 3
Notice!
COMBINATION (C)
=
CHOOSE!

5. Factorial Formula for Combinations
!
.
! !( )!
n r
n r
P n
C
r r n r
Example:
The manager of a football team has a squad of 18 players. He needs to choose 11 to play
in a match. How many possible teams can be choosen?
Answer: Using Factional formula of C,
18C11= = 31824
!7!*11
!18
)!1118(!11
!18
!11
1118P
You can using your scientific
calcuator, it simple, you enter number n, then
shift-( )-then enter the number of r, try it!
18-shift-( )-P- then, 11 so it became to
18C11= 31824

6. Difference between Permutation and Combination

7. Exercises:
1. From a group of 7 men and 6 women, five persons are to be selected to
form a committee so that at least 3 men are there on the committee. In how
many ways can it be done?
A.564 B.645 C.735 D.756 E.None of these
Answer: Option D
Explanation:
We may have (3 men and 2 women) or (4 men and 1 woman) or (5 men only).
Required number of ways= (7C3 x 6C2) + (7C4 x 6C1) + (7C5)=7 x 6 x 5x6 x 5+
(7C3 x 6C1) + (7C2)3 x 2 x 12 x 1= 525 +7 x 6 x 5x 6+7 x 63 x 2 x 12 x 1= (525 + 210
+ 21)= 756.
2. A box contains 2 white balls, 3 black balls and 4 red balls. In how many
ways can 3 balls be drawn from the box, if at least one black ball is to be
included in the draw?
A.32 B.48 C.64 D.96 E.None of these
Answer: Option C
Explanation:
We may have(1 black and 2 non-black) or (2 black and 1 non-black) or (3
black).
Required number of ways= (3C1 x 6C2) + (3C2 x 6C1) + (3C3)=3 x6 x 5+3 x 2x 6+
12 x 12 x 1= (45 + 18 + 1)= 64.

8. 3. In how many different ways can the letters of the word 'LEADING' be
arranged in such a way that the vowels always come together?
A.360 B.480 C.720 D.5040 E.None of these
Answer: Option C
Explanation:
The word 'LEADING' has 7 different letters.
When the vowels EAI are always together, they can be supposed to form one
letter.
Then, we have to arrange the letters LNDG (EAI).
Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways.
The vowels (EAI) can be arranged among themselves in 3! = 6 ways.
Required number of ways = (120 x 6) = 720.
4. In how many ways can the letters of the word 'LEADER' be arranged?
A.72 B.144 C.360 D.720 E.None of these
Answer: Option C
Explanation:
The word 'LEADER' contains 6 letters, namely 1L, 2E, 1A, 1D and 1R.
Required number of ways = = 360
)!1)(!1)(!1)(!2)(!1(
!6

9. Thank you!!