# week2 - S Tanny Subsets of a Set[n 1 How many k-subsets...

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S. Tanny MAT 344F Fall, 1997 Subsets of a Set [n] 1) How many k-subsets of [n] are there? k 0, integer n = 4 12 13 14 k = 2 34 23 24 let x be the # of k-subsets Each such subset can be arranged in k! ways. Thus, x k! counts the number of ordered k-subsets of [n], which is just n k x k! = n k x = k n k! n k What is n 0 ? 0 0 ? 3 4 ? Notice: n + 1 k = n k + n k - 1 (This is called the triangle formula for binomial coefficients.) Fix your eye on the element. ( n + 1): (n + 1) is in or out of any subset n k counts all k-subsets where (n + 1) is OUT (because these are just k-subsets of [n]). n k - 1 counts all k-subsets where (n + 1) is in. By the SUM rule, this counts all k-subsets of [n + 1].

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S. Tanny MAT 344F Fall, 1997 "Algebraic" Proof of the above identity: n k + n k - 1 = n! k!(n - k)! + n! (k - 1)!(n - k + 1)! = = n! (k - 1)!(n - k)! n + 1 k(n - k + 1) = (n + 1)! k!(n - k + 1)! n + 1 k Note: n k = n n - k Each choice of a k-subset leaves behind an (n - k) subset.
S. Tanny MAT 344F Fall, 1997 Graphs of Binomial Coefficients f 2 (n) = n 2 f 3 (n) = n 3 g 2 (k) = 2 k g 3 (k) = 3 k

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S. Tanny MAT 344F Fall, 1997 f 2 (n) = n 2 = n(n - 1) 2 f 3 (n) = n 3 = 1 6 n(n - 1) (n - 2) g 3 (r) = 3 r g 6 (r) = 6 r Unimodal:up/down Single or Double maximum
S. Tanny MAT 344F Fall, 1997 Array of Binomial Coefficients n 0 = 1 n 0 1 1 1 1 2 1 1 3 3 1 1

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