# 12x1 t08 05 binomial coefficients (2010)

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• 1. Relationships Between Binomial Coefficients

2. Relationships BetweenBinomial Coefficients Binomial Theorem 3. Relationships Between Binomial Coefficientsn Binomial Theorem 1 x n nCk x k k 0 4. Relationships Between Binomial Coefficientsn Binomial Theorem 1 x n nCk x k k 0 nC0 nC1 x nC2 x 2 nCk x k nCn x n 5. Relationships BetweenBinomial Coefficients nBinomial Theorem 1 x n n Ck x kk 0 nC0 nC1 x nC2 x 2 nCk x k nCn x n e.g. (i) Find the values of; na) nCkk 1 6. Relationships Between Binomial Coefficients nBinomial Theorem 1 x n n Ck x kk 0 nC0 nC1 x nC2 x 2 nCk x k nCn x n e.g. (i) Find the values of; nna) Ck n1 x nCk x kn k 1 k 0 7. Relationships Between Binomial Coefficients nBinomial Theorem 1 x n n Ck x kk 0 nC0 nC1 x nC2 x 2 nCk x k nCn x n e.g. (i) Find the values of; nna) Ck n1 x nCk x kn k 1 k 0 n let x = 1; 1 1n nCk 1k k 0 8. Relationships Between Binomial Coefficients nBinomial Theorem 1 x n n Ck x kk 0 nC0 nC1 x nC2 x 2 nCk x k nCn x n e.g. (i) Find the values of; nna) Ck n1 x nCk x kn k 1 k 0 n let x = 1; 1 1n nCk 1k k 0 n2 n nCkk 0 9. Relationships Between Binomial Coefficients nBinomial Theorem 1 x n n Ck x kk 0 nC0 nC1 x nC2 x 2 nCk x k nCn x n e.g. (i) Find the values of; nna) Ck n1 x nCk x kn k 1 k 0 n let x = 1; 1 1n nCk 1k k 0 n2 n nCkk 0 n 2 n n C0 n C kk 1 10. Relationships Between Binomial CoefficientsnBinomial 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 na) Ck n n1 x Ck x n nkn C k 2 n n C0k 1k 0k 1n let x = 1; 1 1n nCk 1kk 0n2 n nCk k 0n2 n n C0 n C k k 1 11. Relationships Between Binomial CoefficientsnBinomial 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 na) Ck n n1 x Ck x n nk nC k 2 n n C0k 1k 0k 1nn let x = 1; 1 1n nCk 1k nCk 2 n 1k 0 k 1n2 n nCk k 0n2 n n C0 n C k k 1 12. b) nC1 nC3 nC5 nC7 13. b) nC1 nC3 nC5 nC7 n1 x nCk x kn k 0 14. b) nC1 nC3 nC5 nC7 n1 x nCk x knk 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 1let 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 1let x = -1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5 n0 nC0 nC1 nC2 nC3 nC4 nC5 2 19. b) nC1 nC3 nC5 nC7 n 1 x nCk x knk 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 1let x = -1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5 n0 nC0 nC1 nC2 nC3 nC4 nC5 2 subtract (2) from (1) 20. b) nC1 nC3 nC5 nC7 n 1 x nCk x knk 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 1let x = -1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5 n0 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 knk 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 1let x = -1; 1 1 nC0 nC1 nC2 nC3 nC4 nC5 n0 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 22. n c) k nCk k 1 23. n c) k nCk k 1n1 x nCk x kn k 0 24. n c) k nCk k 1 n 1 x nCk x k nk 0 Differentiate both sides 25. n c) k nCk k 1n 1 x nCk x k n k 0 Differentiate both sides nn1 x k nCk x k 1 n 1k 0 26. n c) k nCk k 1n 1 x nCk x k n k 0 Differentiate both sides nn1 x k nCk x k 1 n 1k 0nn1 1 k nCkn 1 let x = 1; k 0 27. n c) k nCk k 1 n 1 x nCk x knk 0 Differentiate both sidesnn1 x k nCk x k 1n 1 k 0 nn1 1 k nCkn 1 let x = 1;k 0 n n 2 0 C0 k nCk n 1 n k 1 28. n c) k nCk k 1n 1 x nCk x k n k 0 Differentiate both sides nn1 x k nCk x k 1 n 1k 0nn1 1 k nCk n 1 let x = 1; k 0 n n 2 0 C0 k nCk n 1n k 1 n k C n 2 n n 1 kk 1 29. n 1k nCk d) k 0 k 1 30. n 1k nCk d) k 1n 1 x nCk x k k 0nk 0 31. n 1k nCk d) k 1n 1 x nCk x k k 0nk 0 Integrate both sides 32. n 1k nCk d) k 1n 1 x nCk x k k 0nk 0 Integrate both sides1 x n1 n x k 1 K Ck nn 1 k 0 k 1 33. n 1k nCk d) k 1n 1 x nCk x k k 0nk 0Integrate both sides 1 x n1 nx k 1 K nCkn 1 k 0 k 1 1 0n 1 n0 k 1 let x = 0; K nCkn 1 k 0 k 1 34. n 1k nCk d) k 1n 1 x nCk x k k 0nk 0Integrate both sides 1 x n1 nx k 1 K nCkn 1 k 0 k 1 1 0n 1 n0 k 1 let x = 0; K nCkn 1 k 0 k 1 1 K n 1 35. n 1k nCk d) k 1n 1 x nCk x k k 0nk 0Integrate both sides 1 x n1 K n nC x k 1n 1 k k 1 k 01 0n 1 K n nC 0 k 1 let x = 0;n 1 k k 1 k 01K n 11 1n 1 1 n nC 1k 1 k k 1 let x = -1; n 1 k 0 36. n 1k nCk d) k 1n 1 x nCk x k k 0nk 0Integrate both sides 1 x n1 K n nC x k 1 n 1 k k 1k 0 1 0n 1 K n nC 0 k 1 let x = 0; n 1 k k 1k 0 1 Kn 1 1 1n 1 1 n nC 1k 1 k k 1 let x = -1; n 1k 0n 1k 1 1 Ck k 1 n 1k 0 n 37. n 1k nCk d) k 1 n 1 x nCk x k k 0nk 0Integrate both sides 1 x n1 K n nC x k 1 n 1 k k 1k 0 1 0n 1 K n nC 0 k 1 let x = 0; n 1 k k 1k 0 1 Kn 1 1 1n 1 1 n nC 1k 1 k k 1 let x = -1; n 1k 0n 1k 1 1 Ck k 1 n 1k 0 nn 1k 1 Cknk 0 k 1n 1 38. ii By equating the coefficients of x n on both sides of the identity;1 x 1 x 1 x nn2nshow that; n 2n ! n 2 k n!2k 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 kk 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 nn0 coefficient of x n in 1 x 1 x nn 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 nn0 coefficient of x n in 1 x 1 x nn 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 !n2 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 nn0 coefficient of x n in 1 x 1 x nn 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 22 n 11 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 nn0 coefficient of x n in 1 x 1 x nn 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 xx 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 nn0 coefficient of x n in 1 x 1 x nn 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 xx 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 nn0 coefficient of x n in 1 x 1 x nn 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 xx 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 11 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 2n 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 2n 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 2n 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

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