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# 12 x1 t08 05 binomial coefficients (2013)

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