lec18-binomial - EECS 203 Winter 2007 Discrete Mathematics...

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Unformatted text preview: EECS 203, Winter 2007 Discrete Mathematics Lecture 18 Binomial Coefficients March 13 Reading: Rosen [5.4] March 13 Binomial Coefficients, Page 1 18.1 Binomial Theorem n (x + y) = j=0 n n n-j j x y . j March 13 Binomial Coefficients, Page 2 18.2 Properties of the binomial coefficients Pascal's Identity: n+1 k Pascal's triangle. Growth: n n / k k-1 Thus n k = n n + . k k-1 = (n - k + 1)/k. increases then decreases. n m If n = 2m + 1, If n = 2m, n m = n m+1 are maximum; is maximum. Magnitude (for large n and k): n k n/(2k(n - k))2nH(k/n) , March 13 Binomial Coefficients, Page 3 18.2 Properties of the binomial coefficients where 1 1 H() = log + (1 - ) log 1- is the entropy function. n n/2 2 n 2 . n March 13 Binomial Coefficients, Page 4 18.3 Identities involving binomial coefficients n k=0 n n k k = 2n . (-1) k=0 n k = 0. m+n r r = k=0 m r-k n n . k n+1 r+1 = j=r j . r March 13 Binomial Coefficients, Page 5 18.4 Permutations and combinations with repetition A r-permutation with repetition of a set A is an order arrangement of possibly repeating elements from A. Fact. The number of r-permutations with repetitions of a n-element set is nr . A r-combinations with repetitions of a set A is a collection of r possibly repeating elements from A. (That is, a multi-set of r elements, each is from A.) Fact. The number of r-combinations with repetitions of an n-element set is n+r-1 . n-1 Proof. Examples. How many nonnegative integer solutions to x1 + x2 + + xn = r? March 13 Binomial Coefficients, Page 6 18.4 Permutations and combinations with repetition How many choices of numbers p1 , p2 , , pr such that 1 p1 p2 pr n? Shopping March 13 Binomial Coefficients, Page 7 18.5 Permutations with indistinguishable objects The number of different permutations of n objects, where there are ni indistinguishable objects of type i, i = 1, ..., k, is n! . n1 !n2 ! nk ! Examples. March 13 Binomial Coefficients, Page 8 ...
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This note was uploaded on 04/01/2008 for the course EECS 203 taught by Professor Yaoyunshi during the Winter '07 term at University of Michigan.

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lec18-binomial - EECS 203 Winter 2007 Discrete Mathematics...

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