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Binomial Coefficient

Binomial Coefficient. Supplementary Notes. Prepared by Raymond Wong. Presented by Raymond Wong. e.g.1 (Page 4). Prove that. S 1. S 2. S 3. S 4. S 0. {1}. {1, 2, 3}. {1, 2}. {1, 3}. {2}. {1, 2, 3, 4}. {1, 2, 4}. {1, 4}. {2, 3}. {}. {3}. {1, 3, 4}. {2, 4}. {3, 4}. {4}.

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Binomial Coefficient

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  1. Binomial Coefficient Supplementary Notes Prepared by Raymond Wong Presented by Raymond Wong

  2. e.g.1 (Page 4) • Prove that

  3. S1 S2 S3 S4 S0 {1} {1, 2, 3} {1, 2} {1, 3} {2} {1, 2, 3, 4} {1, 2, 4} {1, 4} {2, 3} {} {3} {1, 3, 4} {2, 4} {3, 4} {4} {2, 3, 4} e.g.1 A set of all possible subsets of {1, 2, 3, 4} {1, 2, 3} {1} {1, 2} {1, 3} {1, 2, 4} {2} {1, 4} {2, 3} {1, 2, 3, 4} {} {1, 3, 4} {3} {2, 4} {3, 4} {2, 3, 4} {4}

  4. 4 4 4 3 4 2 4 1 4 0 S1 S2 S3 S4 S0 {1} {1, 2, 3} {1, 2} {1, 3} {2} {1, 2, 3, 4} {1, 2, 4} {1, 4} {2, 3} {} {3} {1, 3, 4} {2, 4} {3, 4} {4} {2, 3, 4} e.g.1 A set of all possible subsets of {1, 2, 3, 4} {1, 2, 3} {1} {1, 2} {1, 3} {1, 2, 4} {2} {1, 4} {2, 3} {1, 2, 3, 4} {} {1, 3, 4} {3} {2, 4} {3, 4} {2, 3, 4} {4} + + + +

  5. {2} {2, 3, 4} e.g.1 A set of all possible subsets of {1, 2, 3, 4} {1, 2, 3} {1} {1, 2} {1, 3} {1, 2, 4} {2} {1, 4} {2, 3} {1, 2, 3, 4} {} {1, 3, 4} {3} {2, 4} {3, 4} {2, 3, 4} {4} 1 does not appear 2 appears 3 does not appear 4 does not appear 1 does not appear 2 appears 3 appears 4 appears We can have another representation (related to “whether an element appears or not”)to represent a subset

  6. 4 3 2 1 Appears Appears Appears Appears {2} Does not appear Does not appear Does not appear Does not appear e.g.1 A set of all possible subsets of {1, 2, 3, 4} {1, 2, 3} {1} {1, 2} {1, 3} {1, 2, 4} {2} {1, 4} {2, 3} {1, 2, 3, 4} {} {1, 3, 4} {3} {2, 4} {3, 4} {2, 3, 4} {4}

  7. 4 3 2 1 Appears Appears Appears Appears Does not appear Does not appear Does not appear Does not appear {2, 3, 4} e.g.1 A set of all possible subsets of {1, 2, 3, 4} {1, 2, 3} {1} {1, 2} {1, 3} {1, 2, 4} {2} {1, 4} {2, 3} {1, 2, 3, 4} {} {1, 3, 4} {3} {2, 4} {3, 4} {2, 3, 4} {4}

  8. 2 1 4 3 Appears Appears Appears Appears Does not appear Does not appear Does not appear Does not appear e.g.1 A set of all possible subsets of {1, 2, 3, 4} {1, 2, 3} {1} {1, 2} {1, 3} {1, 2, 4} {2} {1, 4} {2, 3} {1, 2, 3, 4} {} {1, 3, 4} {3} {2, 4} {3, 4} {2, 3, 4} {4} 2 choices 2 choices 2 choices 2 choices Total number of subsets of {1, 2, 3, 4} = 2 x 2 x 2 x 2 = 24

  9. e.g.2 (Page 16) • Prove that

  10. S A set of all possible 2-subsets of S C A {B, C} {A, B} D {C, E} {A, E} B {B, D} {A, C} E {B, E} {D, E} {A, D} {C, D} 5 2 e.g.2

  11. S A set of all possible 2-subsets of S C A {B, C} {A, B} D {C, E} {A, E} B {B, D} {A, C} E {B, E} {D, E} {A, D} {C, D} A set of all possible 2-subsets of S not containing E A set of all possible 2-subsets of S containing E {B, C} {A, B} {C, E} {A, E} {B, D} {A, C} {B, E} {D, E} {A, D} {C, D} e.g.2

  12. S S’ S’ C A A A D C C B E D D B B A set of all possible 2-subsets of S not containing E 4 2 4 1 A set of all possible 2-subsets of S containing E {B, C} {A, B} {C, E} {A, E} {B, D} {A, C} {B, E} {D, E} {A, D} {C, D} e.g.2 We know that each 2-subset contains E. Since each 2-subset contains 2 elements, the other ONE element comes from {A, B, C, D} A set of all possible 2-subsets of {A, B, C, D} + This proof is an example of a combinatorial proof.

  13. e.g.3 (Page 22) • Prove that

  14. green blue red x x x y y y e.g.3 monomial

  15. Set of y’s (in different colors) y 3 2 y y = e.g.3 Interpretation 1 Two y’s are in different colors. Suppose that we choose 2 elements. These two elements correspond to two y’s in different colors. Coefficient of xy2= No. of ways of choosing 2 y’s from this set

  16. Set of positions in the list 1 3 2 2 3 = e.g.3 Interpretation 2 Each monomial has 3 elements. Each element can be x or y. L1L2L3 Each Li can be x or y. Now, I want to have 2 y’s in this list. These 2 y’s appear in 2 different positions in this list. Suppose that we choose 2 positions from the set. These two positions correspond to the positions that y appears. e.g., {1, 3} means yxy Coefficient of xy2= No. of lists containing 2 y’s

  17. Set of positions in the list 1 2 3 e.g.3 Interpretation 3 L1L2L3 Coefficient of xy2=

  18. Set of positions in the list 1 3 2 3 2 2 3 Choose 2 objects = e.g.3 Interpretation 3 L1L2L3 Bucket B1 Bucket B2 Coefficient of xy2= No. of ways of distributing 3 objects

  19. e.g.4 (Page 29) • Suppose we have 2 distinguishable buckets, namely B1 and B2 • How many ways can we distribute 5 objects into these buckets such that • 2 objects are in B1 • 3 objects are in B2?

  20. Choose 2 objects 5 2 e.g.4 B1 B2

  21. 9 objects  2 objects in B1, 3 objects in B2 and 4 objects in B3 e.g.5 (Page 30) • Suppose we have 3 distinguishable buckets, namely B1, B2 and B3. • How many ways can we distribute 9 objects into these buckets such that • 2 objects are in B1 • 3 objects are in B2, and • 4 objects are in B3?

  22. Choose 2 objects 9 2 9 objects  2 objects in B1, 3 objects in B2 and 4 objects in B3 e.g.5 B1

  23. 9! 7! x x = 2!(9-2)! 3!(7-3)! 9! = 2!3!4! Choose 3 objects Choose 2 objects 7 3 9 2 7 3 9 2 9 objects  2 objects in B1, 3 objects in B2 and 4 objects in B3 e.g.5 Total number of placing objects = B1 B2 B3

  24. e.g.6 (Page 33) • Prove that the coefficient of in (x + y + z)4is equal towhere k1 + k2 + k3 = 4

  25. Set of positions in the list 1 2 3 Choose 2 objects Choose 1 object 4 = 4 2 1 1 e.g.6 (x + y + z)(x + y + z)(x + y + z)(x + y + z) = xxxx + xxxy + xxxz + xxyx + … + zzzy + zzzz L1L2L3 L4 Bucket B1 Bucket B2 Bucket B3 Coefficient of x2yz=

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