L03_BinomialCoefficients_print

L03_BinomialCoefficients_print - 1 COMP170 Discrete...

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Unformatted text preview: 1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 1.3, pp. 19-26 Binomial Coefficients Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen 2 1.3 Binomial Coefficients • Pascal’s Triangle • A Proof using the Sum Principle • The Binomial Theorem • Labeling and Trinomial Coefficients 3 Some properties of Binomial Coefficients • n k = n ! k !( n- k )! • n = 1 only one set of size . • n n = 1 only one set of size n . • n k = n n- k Obvious from equation. Can you think of a simple bijection that explains this? is the number of k-element subsets of an n-element set. 4 Some properties of Binomial Coefficients (cont) n X i =0 n i = 2 n Use Sum Principle Let P = set of all subsets of { 1,2,. . . ,n } S i = set of all i subsets of { 1,2,. . . ,n } ⇒ | P | = n X i =0 | S i | = n X i =0 n i Let L = L 1 L 2 ...L n be a list of size n from { , 1 } If L = set of all such lists ⇒ |L| = 2 n There is a bijection (next page) between L and P so | P | = 2 n and we are done. 5 Let P = set of all subsets of { 1,2,. . . ,n } Let L = L 1 L 2 ...L n be a list of size n from { , 1 } and L = set of all such lists Define the following function f : L → P If L ∈ L then f ( L ) is the set S ⊆ { 1 ,, 2 ,...,n } defined by i ∈ S ⇔ L i = 1 f (10101) = { 1 , 3 , 5 } , f (11101) = { 1 , 2 , 3 , 5 } , f (00000) = ∅ . Ex: n = 5 f is a bijection between L and P (why?) so |L| = | P | Note: L is sometimes called the incidence vector or membership vector associated with L 6 P = { 1 } { 1 , 2 }{ 1 , 3 } { 1 , 2 , 3 } { 1 , 2 , 3 , 4 } {} { 2 } { 1 , 4 }{ 2 , 3 } { 1 , 2 , 4 } { 3 } { 2 , 4 }{ 3 , 4 } { 1 , 3 , 4 } { 4 } { 2 , 3 , 4 } Example: n = 4 , S = { 1 , 2 , 3 , 4 } 7 P = { 1 } { 1 , 2 }{ 1 , 3 } { 1 , 2 , 3 } { 1 , 2 , 3 , 4 } {} { 2 } { 1 , 4 }{ 2 , 3 } { 1 , 2 , 4 } { 3 } { 2 , 4 }{ 3 , 4 } { 1 , 3 , 4 } { 4 } { 2 , 3 , 4 } P = { S , S 1 , S 2 , S 3 , S 4 } Example: n = 4 , S = { 1 , 2 , 3 , 4 } 8 P = { 1 } { 1 , 2 }{ 1 , 3 } { 1 , 2 , 3 } { 1 , 2 , 3 , 4 } {} { 2 } { 1 , 4 }{ 2 , 3 } { 1 , 2 , 4 } { 3 } { 2 , 4 }{ 3 , 4 } { 1 , 3 , 4 } { 4 } { 2 , 3 , 4 } | S | = 4 , | S 1 | = 4 1 , | S 2 | = 4 2 , | S 3 | = 4 3 , | S 4 | = 4 4 Example: n = 4 , S = { 1 , 2 , 3 , 4 } 9 P = { 1 } { 1 , 2 }{ 1 , 3 } { 1 , 2 , 3 } { 1 , 2 , 3 , 4 } {} { 2 } { 1 , 4 }{ 2 , 3 } { 1 , 2 , 4 } { 3 } { 2 , 4 }{ 3 , 4 } { 1 , 3 , 4 } { 4 } { 2 , 3 , 4 }...
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L03_BinomialCoefficients_print - 1 COMP170 Discrete...

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