Then pnr nn r proof for r in the range 1 r

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Unformatted text preview: mutations Corollary: Let n be a positive integer, and r an integer in the range 0 <= r <= n. Then P(n,r) = n!/(n-r)! Proof: For r in the range 1 <= r <= n, this follows from the previous theorem and the fact that n!/(n-r)! = n(n-1) ... (n-r+1). For r=0, we have P(n,0)=1, which equals n!/(n-0)! =n!/n!=1. 26 Permutation Example How many permutations of the letters ABCDEFGH contain the string ABC? Let us regard ABC, D, E, F, G, and H as blocks. Any permutation of these six blocks will yield a valid permutation containing ABC, and there are no others. Therefore, we have 6!=720 permutations of the letters ABCDEFGH that contain ABC as a block. 27 Combinations Let S be a set of n elements. An r-combination of S is a subset of r elements from S. The number of r-combinations of a set S with n elements is denoted by C(n,r) or ￿￿ n . r 28 Number of r-Combinations Theorem: The number of r-combinations of a set with n elements is given by ￿￿ n n! = r (n − r)!r! Proof. We can form all r-permutations of a set with n elements by first choosing an rcombination and then ordering the r elements in all possible ways. Thus, P(n,r)=C(n,r)P(r,r). Hence, C(n,r)=P(n,r)/P(r,r)=n!/(n-r)!/r!/(r-r)!. Since (r-r)!=0!=1, this yields our claim. Binomial Coefficient Identity ￿￿ ￿ ￿ Corollary: We have n n = r n−r Proof: Let S be a set with n elements. Each subset A of S is determined by its complement Ac, which specifies the elements of S that are not contained in A. Therefore, we can use double counting: The number C(n,r) of subsets of cardinality r of S corresponds to the number of “complements of subsets of cardinality r in S”. Since |A|=r iff |Ac|=n-r, the complements of subsets of cardinality r of S correspond to subsets of cardinality n-r of S. Thus, C(n,r) = # of r-subsets of S = # of complements of r-subsets of S = C(n,n-r), as claimed. 30 Counting Subset Identity Theorem: For any nonnegative integer n, we have n￿ ￿ k=0 n k ￿ n =2 Proof: Let S be a set...
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This note was uploaded on 03/24/2014 for the course CSCE 222 taught by Professor Math during the Fall '11 term at Texas A&M.

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