lec18-binomial

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

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EECS 203, Winter 2007 Discrete Mathematics Lecture 18 Binomial Coefficients March 13 Reading: Rosen [5.4] March 13 Binomial Coefficients, Page 1

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18.1 Binomial Theorem ( x + y ) n = n j =0 n j x n - j y j . March 13 Binomial Coefficients, Page 2
18.2 Properties of the binomial coefficients Pascal’s Identity: n + 1 k = n k + n k - 1 . Pascal’s triangle. Growth: n k / n k - 1 = ( n - k + 1) /k. Thus ( n k ) increases then decreases. If n = 2 m + 1, ( n m ) = ( n m +1 ) are maximum; If n = 2 m , ( n m ) is maximum. Magnitude (for large n and k ): n k n/ (2 πk ( n - k ))2 nH ( k/n ) , March 13 Binomial Coefficients, Page 3

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18.2 Properties of the binomial coefficients where H ( α ) = α log 1 α + (1 - α ) log 1 1 - α is the entropy function. n n/ 2 2 πn 2 n . March 13 Binomial Coefficients, Page 4
18.3 Identities involving binomial coefficients n k =0 n k = 2 n . n k =0 ( - 1) k n k = 0 . m + n r = r k =0 m r - k n k . n + 1 r + 1 = n j = r j r . March 13 Binomial Coefficients, Page 5

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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 n r . A r

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• Winter '07
• YaoyunShi
• binomial coefficients

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