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12 x1 t08 05 binomial coefficients (2013)

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12 x1 t08 05 binomial coefficients (2013)

1. 1. Relationships BetweenBinomial Coefficients
2. 2. Relationships BetweenBinomial CoefficientsBinomial Theorem
3. 3. Relationships BetweenBinomial CoefficientsBinomial Theorem   nkkknnxCx01
4. 4. Relationships BetweenBinomial CoefficientsBinomial Theorem   nkkknnxCx01nnnkknnnnxCxCxCxCC  2210
5. 5. Relationships BetweenBinomial CoefficientsBinomial Theorem   nkkknnxCx01nnnkknnnnxCxCxCxCC  2210e.g. (i) Find the values of;nkknC1a)
6. 6. Relationships BetweenBinomial CoefficientsBinomial Theorem   nkkknnxCx01nnnkknnnnxCxCxCxCC  2210e.g. (i) Find the values of;nkknC1a)   nkkknnxCx01
7. 7. Relationships BetweenBinomial CoefficientsBinomial Theorem   nkkknnxCx01nnnkknnnnxCxCxCxCC  2210e.g. (i) Find the values of;nkknC1a)let x = 1;  nkkknnxCx01  nkkknnC0111
8. 8. Relationships BetweenBinomial CoefficientsBinomial Theorem   nkkknnxCx01nnnkknnnnxCxCxCxCC  2210e.g. (i) Find the values of;nkknC1a)let x = 1;  nkkknnxCx01  nkkknnC0111nkknnC02
9. 9. Relationships BetweenBinomial CoefficientsBinomial Theorem   nkkknnxCx01nnnkknnnnxCxCxCxCC  2210e.g. (i) Find the values of;nkknC1a)let x = 1;  nkkknnxCx01  nkkknnC0111nkknnC02nkknnnCC102
10. 10. Relationships BetweenBinomial CoefficientsBinomial Theorem   nkkknnxCx01nnnkknnnnxCxCxCxCC  2210e.g. (i) Find the values of;nkknC1a)let x = 1;  nkkknnxCx01  nkkknnC0111nkknnC02nkknnnCC102012 CC nnnkkn
11. 11. Relationships BetweenBinomial CoefficientsBinomial Theorem   nkkknnxCx01nnnkknnnnxCxCxCxCC  2210e.g. (i) Find the values of;nkknC1a)let x = 1;  nkkknnxCx01  nkkknnC0111nkknnC02nkknnnCC102012 CC nnnkkn121nnkknC
12. 12.  7531b) CCCC nnnn
13. 13.  7531b) CCCC nnnn  nkkknnxCx01
14. 14.  7531b) CCCC nnnn  nkkknnxCx01 5544332210 xCxCxCxCxCC nnnnnn
15. 15.  7531b) CCCC nnnnlet x = 1;  nkkknnxCx01 5544332210 xCxCxCxCxCC nnnnnn   54321011 CCCCCC nnnnnnn
16. 16.  7531b) CCCC nnnnlet x = 1;  nkkknnxCx01 5544332210 xCxCxCxCxCC nnnnnn   54321011 CCCCCC nnnnnnn 12 543210  CCCCCC nnnnnnn
17. 17.  7531b) CCCC nnnnlet x = 1;  nkkknnxCx01 5544332210 xCxCxCxCxCC nnnnnn   54321011 CCCCCC nnnnnnn 12 543210  CCCCCC nnnnnnnlet x = -1;    54321011 CCCCCC nnnnnnn
18. 18.  7531b) CCCC nnnnlet x = 1;  nkkknnxCx01 5544332210 xCxCxCxCxCC nnnnnn   54321011 CCCCCC nnnnnnn 12 543210  CCCCCC nnnnnnnlet x = -1;    54321011 CCCCCC nnnnnnn 20 543210  CCCCCC nnnnnn
19. 19.  7531b) CCCC nnnnlet x = 1;  nkkknnxCx01 5544332210 xCxCxCxCxCC nnnnnn   54321011 CCCCCC nnnnnnn 12 543210  CCCCCC nnnnnnnlet x = -1;    54321011 CCCCCC nnnnnnn 20 543210  CCCCCC nnnnnnsubtract (2) from (1)
20. 20.  7531b) CCCC nnnnlet x = 1;  nkkknnxCx01 5544332210 xCxCxCxCxCC nnnnnn   54321011 CCCCCC nnnnnnn 12 543210  CCCCCC nnnnnnnlet x = -1;    54321011 CCCCCC nnnnnnn 20 543210  CCCCCC nnnnnnsubtract (2) from (1) 531 2222 CCC nnnn
21. 21.  7531b) CCCC nnnnlet x = 1;  nkkknnxCx01 5544332210 xCxCxCxCxCC nnnnnn   54321011 CCCCCC nnnnnnn 12 543210  CCCCCC nnnnnnnlet x = -1;    54321011 CCCCCC nnnnnnn 20 543210  CCCCCC nnnnnnsubtract (2) from (1) 531 2222 CCC nnnn53112 CCC nnnn
22. 22. nkknCk1c)
23. 23. nkknCk1c)  nkkknnxCx01
24. 24. nkknCk1c)Differentiate both sides  nkkknnxCx01
25. 25. nkknCk1c)Differentiate both sides  nkkknnxCx01  1011  knkknnxCkxn
26. 26. nkknCk1c)Differentiate both sides  nkkknnxCx01  1011  knkknnxCkxnlet x = 1;   nkknnCkn0111
27. 27. nkknCk1c)Differentiate both sides  nkkknnxCx01  1011  knkknnxCkxnlet x = 1;   nkknnCkn0111    nkknnnCkCn10102
28. 28. nkknCk1c)Differentiate both sides  nkkknnxCx01  1011  knkknnxCkxnlet x = 1;   nkknnCkn0111    nkknnnCkCn10102  112nnkknnCk
29. 29.   nkknkkC0 11d)
30. 30.   nkknkkC0 11d)  nkkknnxCx01
31. 31.   nkknkkC0 11d)Integrate both sides  nkkknnxCx01
32. 32.   nkknkkC0 11d)Integrate both sides  nkkknnxCx01 111 101  kxCKnx knkknn
33. 33.   nkknkkC0 11d)Integrate both sides  nkkknnxCx01 111 101  kxCKnx knkknnlet x = 0; 10101 101  kCKnknkknn
34. 34.   nkknkkC0 11d)Integrate both sides  nkkknnxCx01 111 101  kxCKnx knkknnlet x = 0; 10101 101  kCKnknkknn11nK
35. 35.   nkknkkC0 11d)Integrate both sides  nkkknnxCx01 111 101  kxCKnx knkknnlet x = 0; 10101 101  kCKnknkknn11nKlet x = -1;    111111101 kCnknkknn
36. 36.   nkknkkC0 11d)Integrate both sides  nkkknnxCx01 111 101  kxCKnx knkknnlet x = 0; 10101 101  kCKnknkknn11nKlet x = -1;    111111101 kCnknkknn 111110  nkCknkkn
37. 37.   nkknkkC0 11d)Integrate both sides  nkkknnxCx01 111 101  kxCKnx knkknnlet x = 0; 10101 101  kCKnknkknn11nKlet x = -1;    111111101 kCnknkknn 111110  nkCknkkn 11110  nkCknkkn
38. 38.        nnnnxxxxii2111identity;theofsidesbothonoftscoefficientheequatingBy  220 !!2that;shownnknnk
39. 39.        nnnnxxxxii2111identity;theofsidesbothonoftscoefficientheequatingBy  220 !!2that;shownnknnk  nkkknnxCx01
40. 40.        nnnnxxxxii2111identity;theofsidesbothonoftscoefficientheequatingBy  220 !!2that;shownnknnk  nkkknnxCx01nnnnnnnnnnnnxCxCxCxCxCC  11222210 
41. 41.        nnnnxxxxii2111identity;theofsidesbothonoftscoefficientheequatingBy  220 !!2that;shownnknnk  nkkknnxCx01nnnnnnnnnnnnxCxCxCxCxCC  11222210    nnnxxx  11inoftcoefficien
42. 42.        nnnnxxxxii2111identity;theofsidesbothonoftscoefficientheequatingBy  220 !!2that;shownnknnk  nkkknnxCx01nnnnnnnnnnnnxCxCxCxCxCC  11222210    nnnxxx  11inoftcoefficien  nnnnnnnnnnnnnnnnnnnnnnnnxCxCxCxCxCCxCxCxCxCxCC1122221011222210
43. 43.        nnnnxxxxii2111identity;theofsidesbothonoftscoefficientheequatingBy  220 !!2that;shownnknnk  nkkknnxCx01nnnnnnnnnnnnxCxCxCxCxCC  11222210    nnnxxx  11inoftcoefficiennxnnn0 01122122 nxnnxnxnnxnxnn nnn  nnnnnnnnnnnnnnnnnnnnnnnnxCxCxCxCxCCxCxCxCxCxCC1122221011222210
44. 44.        nnnnxxxxii2111identity;theofsidesbothonoftscoefficientheequatingBy  220 !!2that;shownnknnk  nkkknnxCx01nnnnnnnnnnnnxCxCxCxCxCC  11222210    nnnxxx  11inoftcoefficiennxnnn0111 nxnnxn  nnnnnnnnnnnnnnnnnnnnnnnnxCxCxCxCxCCxCxCxCxCxCC1122221011222210
45. 45.        nnnnxxxxii2111identity;theofsidesbothonoftscoefficientheequatingBy  220 !!2that;shownnknnk  nkkknnxCx01nnnnnnnnnnnnxCxCxCxCxCC  11222210    nnnxxx  11inoftcoefficiennxnnn0111 nxnnxn 2222 nxnnxn  nnnnnnnnnnnnnnnnnnnnnnnnxCxCxCxCxCCxCxCxCxCxCC1122221011222210
46. 46.        nnnnxxxxii2111identity;theofsidesbothonoftscoefficientheequatingBy  220 !!2that;shownnknnk  nkkknnxCx01nnnnnnnnnnnnxCxCxCxCxCC  11222210    nnnxxx  11inoftcoefficiennxnnn0111 nxnnxn 2222 nxnnxn 01122122 nxnnxnxnnxnxnn nnn  nnnnnnnnnnnnnnnnnnnnnnnnxCxCxCxCxCCxCxCxCxCxCC1122221011222210
47. 47. 022110oftcoefficiennnnnnnnnnnnnxn
48. 48. knnknBut022110oftcoefficiennnnnnnnnnnnnxn
49. 49. knnknBut2222210nnnnn022110oftcoefficiennnnnnnnnnnnnxn
50. 50. knnknBut2222210nnnnnnk kn02022110oftcoefficiennnnnnnnnnnnnxn
51. 51. knnknBut2222210nnnnnnk kn02  nnxx21inoftcoefficien 022110oftcoefficiennnnnnnnnnnnnxn
52. 52. knnknBut2222210nnnnnnk kn02  nnxx21inoftcoefficien   nnnxnnxnnxnxnnx 2222222212021  022110oftcoefficiennnnnnnnnnnnnxn
53. 53. knnknBut2222210nnnnnnk kn02  nnxx21inoftcoefficien   nnnxnnxnnxnxnnx 2222222212021  022110oftcoefficiennnnnnnnnnnnnxn
54. 54. knnknBut2222210nnnnnnk kn02  nnxx21inoftcoefficien   nnnxnnxnnxnxnnx 2222222212021  022110oftcoefficiennnnnnnnnnnnnxnnnxn 2oftcoefficien
55. 55. Now      nnnxxx2111 
56. 56. Now      nnnxxx2111  nnknnk202
57. 57. Now      nnnxxx2111  nnknnk202 !!!2nnn
58. 58. Now      nnnxxx2111  nnknnk202 !!!2nnn  2!!2nn
59. 59. Now      nnnxxx2111  nnknnk202 !!!2nnn  2!!2nnExercise 5F;4, 5, 6, 8, 10,15+ worksheets