3. Relationships Between
Binomial Coefficients
n
Binomial Theorem 1 x n
n
Ck x k
k 0
4. Relationships Between
Binomial Coefficients
n
Binomial Theorem 1 x n
n
Ck x k
k 0
nC0 nC1 x nC2 x 2 nCk x k nCn x n
5. Relationships Between
Binomial Coefficients
n
Binomial Theorem 1 x n
n
Ck x k
k 0
nC0 nC1 x nC2 x 2 nCk x k nCn x n
e.g. (i) Find the values of;
n
a) nCk
k 1
6. Relationships Between
Binomial Coefficients
n
Binomial Theorem 1 x n
n
Ck x k
k 0
nC0 nC1 x nC2 x 2 nCk x k nCn x n
e.g. (i) Find the values of;
n n
a) Ck n
1 x nCk x k
n
k 1 k 0
7. Relationships Between
Binomial Coefficients
n
Binomial Theorem 1 x n
n
Ck x k
k 0
nC0 nC1 x nC2 x 2 nCk x k nCn x n
e.g. (i) Find the values of;
n n
a) Ck n
1 x nCk x k
n
k 1 k 0
n
let x = 1; 1 1n nCk 1k
k 0
8. Relationships Between
Binomial Coefficients
n
Binomial Theorem 1 x n
n
Ck x k
k 0
nC0 nC1 x nC2 x 2 nCk x k nCn x n
e.g. (i) Find the values of;
n n
a) Ck n
1 x nCk x k
n
k 1 k 0
n
let x = 1; 1 1n nCk 1k
k 0
n
2 n nCk
k 0
9. Relationships Between
Binomial Coefficients
n
Binomial Theorem 1 x n
n
Ck x k
k 0
nC0 nC1 x nC2 x 2 nCk x k nCn x n
e.g. (i) Find the values of;
n n
a) Ck n
1 x nCk x k
n
k 1 k 0
n
let x = 1; 1 1n nCk 1k
k 0
n
2 n nCk
k 0
n
2 n n C0 n C k
k 1
10. Relationships Between
Binomial Coefficients
n
Binomial Theorem 1 x n
n
Ck x k
k 0
nC0 nC1 x nC2 x 2 nCk x k nCn x n
e.g. (i) Find the values of;
n n
a) Ck
n
n
1 x Ck x
n n k
n
C k 2 n n C0
k 1 k 0 k 1
n
let x = 1; 1 1n nCk 1k
k 0
n
2 n nCk
k 0
n
2 n n C0 n C k
k 1
11. Relationships Between
Binomial Coefficients
n
Binomial Theorem 1 x n
n
Ck x k
k 0
nC0 nC1 x nC2 x 2 nCk x k nCn x n
e.g. (i) Find the values of;
n n
a) Ck
n
n
1 x Ck x
n n k
n
C k 2 n n C0
k 1 k 0 k 1
n n
let x = 1; 1 1n nCk 1k n
Ck 2 n 1
k 0 k 1
n
2 n nCk
k 0
n
2 n n C0 n C k
k 1
13. b) nC1 nC3 nC5 nC7
n
1 x nCk x k
n
k 0
14. b) nC1 nC3 nC5 nC7
n
1 x nCk x k
n
k 0
nC0 nC1 x nC2 x 2 nC3 x 3 nC4 x 4 nC5 x 5
15. b) nC1 nC3 nC5 nC7
n
1 x nCk x k
n
k 0
nC0 nC1 x nC2 x 2 nC3 x 3 nC4 x 4 nC5 x 5
let x = 1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
16. b) nC1 nC3 nC5 nC7
n
1 x nCk x k
n
k 0
nC0 nC1 x nC2 x 2 nC3 x 3 nC4 x 4 nC5 x 5
let x = 1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
2 n nC0 nC1 nC2 nC3 nC4 nC5 1
17. b) nC1 nC3 nC5 nC7
n
1 x nCk x k
n
k 0
nC0 nC1 x nC2 x 2 nC3 x 3 nC4 x 4 nC5 x 5
let x = 1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
2 n nC0 nC1 nC2 nC3 nC4 nC5 1
let x = -1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
18. b) nC1 nC3 nC5 nC7
n
1 x nCk x k
n
k 0
nC0 nC1 x nC2 x 2 nC3 x 3 nC4 x 4 nC5 x 5
let x = 1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
2 n nC0 nC1 nC2 nC3 nC4 nC5 1
let x = -1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
0 nC0 nC1 nC2 nC3 nC4 nC5 2
19. b) nC1 nC3 nC5 nC7
n
1 x nCk x k
n
k 0
nC0 nC1 x nC2 x 2 nC3 x 3 nC4 x 4 nC5 x 5
let x = 1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
2 n nC0 nC1 nC2 nC3 nC4 nC5 1
let x = -1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
0 nC0 nC1 nC2 nC3 nC4 nC5 2
subtract (2) from (1)
20. b) nC1 nC3 nC5 nC7
n
1 x nCk x k
n
k 0
nC0 nC1 x nC2 x 2 nC3 x 3 nC4 x 4 nC5 x 5
let x = 1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
2 n nC0 nC1 nC2 nC3 nC4 nC5 1
let x = -1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
0 nC0 nC1 nC2 nC3 nC4 nC5 2
subtract (2) from (1)
2 n 2 n C1 2 n C3 2 n C5
21. b) nC1 nC3 nC5 nC7
n
1 x nCk x k
n
k 0
nC0 nC1 x nC2 x 2 nC3 x 3 nC4 x 4 nC5 x 5
let x = 1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
2 n nC0 nC1 nC2 nC3 nC4 nC5 1
let x = -1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5
n
0 nC0 nC1 nC2 nC3 nC4 nC5 2
subtract (2) from (1)
2 n 2 n C1 2 n C3 2 n C5
2 n 1 nC1 nC3 nC5
23. n
c) k nCk
k 1 n
1 x nCk x k
n
k 0
24. n
c) k nCk
k 1 n
1 x nCk x k
n
k 0
Differentiate both sides
25. n
c) k nCk
k 1 n
1 x nCk x k
n
k 0
Differentiate both sides
n
n1 x k nCk x k 1
n 1
k 0
26. n
c) k nCk
k 1 n
1 x nCk x k
n
k 0
Differentiate both sides
n
n1 x k nCk x k 1
n 1
k 0
n
n1 1 k nCk
n 1
let x = 1;
k 0
27. n
c) k nCk
k 1 n
1 x nCk x k
n
k 0
Differentiate both sides
n
n1 x k nCk x k 1
n 1
k 0
n
n1 1 k nCk
n 1
let x = 1;
k 0
n
n 2 0 C0 k nCk
n 1 n
k 1
28. n
c) k nCk
k 1 n
1 x nCk x k
n
k 0
Differentiate both sides
n
n1 x k nCk x k 1
n 1
k 0
n
n1 1 k nCk
n 1
let x = 1;
k 0
n
n 2 0 C0 k nCk
n 1 n
k 1
n
k C n 2
n n 1
k
k 1
30. n
1k nCk
d)
k 1 n
1 x nCk x k
k 0 n
k 0
31. n
1k nCk
d)
k 1 n
1 x nCk x k
k 0 n
k 0
Integrate both sides
32. n
1k nCk
d)
k 1 n
1 x nCk x k
k 0 n
k 0
Integrate both sides
1 x n1 n
x k 1
K Ck n
n 1 k 0 k 1
33. n
1k nCk
d)
k 1 n
1 x nCk x k
k 0 n
k 0
Integrate both sides
1 x n1 n
x k 1
K nCk
n 1 k 0 k 1
1 0n 1 n
0 k 1
let x = 0; K nCk
n 1 k 0 k 1
34. n
1k nCk
d)
k 1 n
1 x nCk x k
k 0 n
k 0
Integrate both sides
1 x n1 n
x k 1
K nCk
n 1 k 0 k 1
1 0n 1 n
0 k 1
let x = 0; K nCk
n 1 k 0 k 1
1
K
n 1
35. n
1k nCk
d)
k 1 n
1 x nCk x k
k 0 n
k 0
Integrate both sides
1 x n1 K n nC x k 1
n 1
k k 1
k 0
1 0n 1 K n nC 0 k 1
let x = 0;
n 1
k k 1
k 0
1
K
n 1
1 1n 1 1 n nC 1k 1
k k 1
let x = -1;
n 1 k 0
36. n
1k nCk
d)
k 1 n
1 x nCk x k
k 0 n
k 0
Integrate both sides
1 x n1 K n nC x k 1
n 1
k k 1
k 0
1 0n 1 K n nC 0 k 1
let x = 0;
n 1
k k 1
k 0
1
K
n 1
1 1n 1 1 n nC 1k 1
k k 1
let x = -1;
n 1 k 0
n
1k 1 1
Ck k 1 n 1
k 0
n
37. n
1k nCk
d)
k 1 n
1 x nCk x k
k 0 n
k 0
Integrate both sides
1 x n1 K n nC x k 1
n 1
k k 1
k 0
1 0n 1 K n nC 0 k 1
let x = 0;
n 1
k k 1
k 0
1
K
n 1
1 1n 1 1 n nC 1k 1
k k 1
let x = -1;
n 1 k 0
n
1k 1 1
Ck k 1 n 1
k 0
n
n
1k 1
Ck
n
k 0 k 1 n 1
38. ii By equating the coefficients of x n on both sides of the identity;
1 x 1 x 1 x
n n 2n
show that;
n 2n !
n 2
k n!2
k 0
39. ii By equating the coefficients of x n on both sides of the identity;
1 x n 1 x n 1 x 2 n
show that;
n 2n !
n 2
k n!2
n k 0
1 x n nCk x k
k 0
40. ii By equating the coefficients of x n on both sides of the identity;
1 x n 1 x n 1 x 2 n
show that;
n 2n !
n 2
k n!2
n k 0
1 x n nCk x k
k 0
C nC1 x nC2 x 2 nCn 2 x n 2 nCn 1 x n 1 nCn x n
n
0
41. ii By equating the coefficients of x n on both sides of the identity;
1 x n 1 x n 1 x 2 n
show that;
n 2n !
n 2
k n!2
n k 0
1 x n nCk x k
k 0
C nC1 x nC2 x 2 nCn 2 x n 2 nCn 1 x n 1 nCn x n
n
0
coefficient of x n in 1 x 1 x
n n
42. ii By equating the coefficients of x n on both sides of the identity;
1 x n 1 x n 1 x 2 n
show that;
n 2n !
n 2
k n!2
n k 0
1 x n nCk x k
k 0
C nC1 x nC2 x 2 nCn 2 x n 2 nCn 1 x n 1 nCn x n
n
0
coefficient of x n in 1 x 1 x
n n
nC0 nC1 x nC2 x 2 nCn2 x n2 nCn1 x n1 nCn x n
nC0 nC1 x nC2 x 2 nCn2 x n2 nCn1 x n1 nCn x n
43. ii By equating the coefficients of x n on both sides of the identity;
1 x n 1 x n 1 x 2 n
show that;
n 2n !
n 2
k n!2
n k 0
1 x n nCk x k
k 0
C nC1 x nC2 x 2 nCn 2 x n 2 nCn 1 x n 1 nCn x n
n
0
coefficient of x n in 1 x 1 x
n n
nC0 nC1 x nC2 x 2 nCn2 x n2 nCn1 x n1 nCn x n
nC0 nC1 x nC2 x 2 nCn2 x n2 nCn1 x n1 nCn x n
n n n
x
0 n
n n2 n 2 n n1 n n n n
x x x x x
n 2 2 n 1 1 n 0
44. ii By equating the coefficients of x n on both sides of the identity;
1 x n 1 x n 1 x 2 n
show that;
n 2n !
n 2
k n!2
n k 0
1 x n nCk x k
k 0
C nC1 x nC2 x 2 nCn 2 x n 2 nCn 1 x n 1 nCn x n
n
0
coefficient of x n in 1 x 1 x
n n
nC0 nC1 x nC2 x 2 nCn2 x n2 nCn1 x n1 nCn x n
nC0 nC1 x nC2 x 2 nCn2 x n2 nCn1 x n1 nCn x n
n n n n n n1
x x x
0 n 1 n 1
45. ii By equating the coefficients of x n on both sides of the identity;
1 x n 1 x n 1 x 2 n
show that;
n 2n !
n 2
k n!2
n k 0
1 x n nCk x k
k 0
C nC1 x nC2 x 2 nCn 2 x n 2 nCn 1 x n 1 nCn x n
n
0
coefficient of x n in 1 x 1 x
n n
nC0 nC1 x nC2 x 2 nCn2 x n2 nCn1 x n1 nCn x n
nC0 nC1 x nC2 x 2 nCn2 x n2 nCn1 x n1 nCn x n
n n n n n n1 n 2 n n2
x x x x x
0 n 1 n 1 2 n 2
46. ii By equating the coefficients of x n on both sides of the identity;
1 x n 1 x n 1 x 2 n
show that;
n 2n !
n 2
k n!2
n k 0
1 x n nCk x k
k 0
C nC1 x nC2 x 2 nCn 2 x n 2 nCn 1 x n 1 nCn x n
n
0
coefficient of x n in 1 x 1 x
n n
nC0 nC1 x nC2 x 2 nCn2 x n2 nCn1 x n1 nCn x n
nC0 nC1 x nC2 x 2 nCn2 x n2 nCn1 x n1 nCn x n
n n n n n n1 n 2 n n2
x x x x x
0 n 1 n 1 2 n 2
n n2 n 2 n n1 n n n n
x x x x x
n 2 2 n 1 1 n 0
47. n n n n n n n n
coefficient of x
n
0 n 1 n 1 2 n 2 n 0
48. n n n n n n n n
coefficient of x
n
0 n 1 n 1 2 n 2 n 0
n n
But
k n k
49. n n n n n n n n
coefficient of x
n
0 n 1 n 1 2 n 2 n 0
n n
But
k n k 2 2 2 2
n n n n
0 1 2 n
50. n n n n n n n n
coefficient of x
n
0 n 1 n 1 2 n 2 n 0
n n
But
k n k 2 2 2 2
n n n n
0 1 2 n
2
n
n
k 0 k
51. n n n n n n n n
coefficient of x
n
0 n 1 n 1 2 n 2 n 0
n n
But
k n k 2 2 2 2
n n n n
0 1 2 n
2
n
n
k 0 k
coefficient of x n in 1 x
2n
52. n n n n n n n n
coefficient of x
n
0 n 1 n 1 2 n 2 n 0
n n
But
k n k 2 2 2 2
n n n n
0 1 2 n
2
n
n
k 0 k
coefficient of x n in 1 x
2n
2n 2n 2n 2 2n n 2n 2 n
1 x
2n
x x x x
0 1 2 n 2n
53. n n n n n n n n
coefficient of x
n
0 n 1 n 1 2 n 2 n 0
n n
But
k n k 2 2 2 2
n n n n
0 1 2 n
2
n
n
k 0 k
coefficient of x n in 1 x
2n
2n 2n 2n 2 2n n 2n 2 n
1 x
2n
x x x x
0 1 2 n 2n
54. n n n n n n n n
coefficient of x
n
0 n 1 n 1 2 n 2 n 0
n n
But
k n k 2 2 2 2
n n n n
0 1 2 n
2
n
n
k 0 k
coefficient of x n in 1 x
2n
2n 2n 2n 2 2n n 2n 2 n
1 x
2n
x x x x
0 1 2 n 2n
2n
coefficient of x
n
n