• Save
12 x1 t08 05 binomial coefficients (2013)
Upcoming SlideShare
Loading in...5
×
 

Like this? Share it with your network

Share

12 x1 t08 05 binomial coefficients (2013)

on

  • 768 views

 

Statistics

Views

Total Views
768
Views on SlideShare
457
Embed Views
311

Actions

Likes
0
Downloads
0
Comments
0

2 Embeds 311

http://virtualb15.edublogs.org 310
http://webcache.googleusercontent.com 1

Accessibility

Categories

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

12 x1 t08 05 binomial coefficients (2013) Presentation Transcript

  • 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