3. General Expansion of
Binomials
Ck is the coefficient of x k in 1 x
n k
n
n
Ck
k
4. General Expansion of
Binomials
Ck is the coefficient of x k in 1 x
n k
n
n
Ck
k
1 x n nC0 nC1 x nC2 x 2 nCn x n
5. General Expansion of
Binomials
Ck is the coefficient of x k in 1 x
n k
n
n
Ck
k
1 x n nC0 nC1 x nC2 x 2 nCn x n
which extends to;
a b n nC0 a n nC1a n1b nC2 a n2b 2 nCn1ab n1 nCnb n
6. General Expansion of
Binomials
Ck is the coefficient of x k in 1 x
n k
n
n
Ck
k
1 x n nC0 nC1 x nC2 x 2 nCn x n
which extends to;
a b n nC0 a n nC1a n1b nC2 a n2b 2 nCn1ab n1 nCnb n
e.g .2 3 x
4
7. General Expansion of
Binomials
Ck is the coefficient of x k in 1 x
n k
n
n
Ck
k
1 x n nC0 nC1 x nC2 x 2 nCn x n
which extends to;
a b n nC0 a n nC1a n1b nC2 a n2b 2 nCn1ab n1 nCnb n
e.g .2 3 x 4C0 2 4 4C1 23 3 x 4C2 2 2 3 x 4C3 23 x 4C4 3 x
4 2 3 4
8. General Expansion of
Binomials
Ck is the coefficient of x k in 1 x
n k
n
n
Ck
k
1 x n nC0 nC1 x nC2 x 2 nCn x n
which extends to;
a b n nC0 a n nC1a n1b nC2 a n2b 2 nCn1ab n1 nCnb n
e.g .2 3 x 4C0 2 4 4C1 23 3 x 4C2 2 2 3 x 4C3 23 x 4C4 3 x
4 2 3 4
16 96 x 216 x 2 216 x 3 81x 4
12. Pascal’s Triangle Relationships
1 nCk n1Ck 1 n1Ck where 1 k n 1
1 x n 1 x 1 x n1
1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1
13. Pascal’s Triangle Relationships
1 nCk n1Ck 1 n1Ck where 1 k n 1
1 x n 1 x 1 x n1
1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1
looking at coefficients of x k
14. Pascal’s Triangle Relationships
1 nCk n1Ck 1 n1Ck where 1 k n 1
1 x n 1 x 1 x n1
1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1
looking at coefficients of x k
LHS nCk
15. Pascal’s Triangle Relationships
1 nCk n1Ck 1 n1Ck where 1 k n 1
1 x n 1 x 1 x n1
1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1
looking at coefficients of x k
LHS nCk
16. Pascal’s Triangle Relationships
1 nCk n1Ck 1 n1Ck where 1 k n 1
1 x n 1 x 1 x n1
1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1
looking at coefficients of x k
LHS nCk RHS 1 n1Ck 1 1 n1Ck
17. Pascal’s Triangle Relationships
1 nCk n1Ck 1 n1Ck where 1 k n 1
1 x n 1 x 1 x n1
1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1
looking at coefficients of x k
LHS nCk RHS 1 n1Ck 1 1 n1Ck
n1Ck 1 n1Ck
18. Pascal’s Triangle Relationships
1 nCk n1Ck 1 n1Ck where 1 k n 1
1 x n 1 x 1 x n1
1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1
looking at coefficients of x k
LHS nCk RHS 1 n1Ck 1 1 n1Ck
n1Ck 1 n1Ck n Ck n1Ck 1 n1Ck
19. Pascal’s Triangle Relationships
1 nCk n1Ck 1 n1Ck where 1 k n 1
1 x n 1 x 1 x n1
1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1
looking at coefficients of x k
LHS nCk RHS 1 n1Ck 1 1 n1Ck
n1Ck 1 n1Ck n Ck n1Ck 1 n1Ck
2 nCk nCnk where 1 k n 1
" Pascal' s triangle is symmetrical"
20. Pascal’s Triangle Relationships
1 nCk n1Ck 1 n1Ck where 1 k n 1
1 x n 1 x 1 x n1
1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1
looking at coefficients of x k
LHS nCk RHS 1 n1Ck 1 1 n1Ck
n1Ck 1 n1Ck n Ck n1Ck 1 n1Ck
2 nCk nCnk where 1 k n 1
" Pascal' s triangle is symmetrical"
3 nC0 nCn 1
21. Pascal’s Triangle Relationships
1 nCk n1Ck 1 n1Ck where 1 k n 1
1 x n 1 x 1 x n1
1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1
looking at coefficients of x k
LHS nCk RHS 1 n1Ck 1 1 n1Ck
n1Ck 1 n1Ck n Ck n1Ck 1 n1Ck
2 nCk nCnk where 1 k n 1
" Pascal' s triangle is symmetrical" Exercise 5B; 2ace, 5, 6ac,
10ac, 11, 14
3 nC0 nCn 1