hw1 - D n,m = D n-1,m D n,m-1(5 How many permutations are...

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HOMEWORK 1 (1) Let S be a set of integers { s 0 ,s 1 ,...,s k } and assume that s 0 >s 1 > · · · > s k . Let M ( S ) = k i =0 ( - 1) i s i . So M ( { 1 , 2 , 5 , 6 , 9 } ) = 7, M ( { 3 } ) = 3, and M ( ) = 0. Compute summationdisplay S [7] M ( S ) where [7] = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . (2) Prove the following identities. Combinatorial proofs are preferred. (a) ( m k ) = m k ( m - 1 k - 1 ) (b) k i =0 ( n i )( r k - i ) = ( n + r k ) (c) n i =0 ( n i ) 2 = ( 2 n n ) (3) Count the number of 5-element subsets of { 1 , 2 ,...,n } that contain a pair of consecutive integers. (4) Let D ( n,m ) denote the number of m -combinations of a multiset with ob- jects of n types each with infinite repetition number. Prove that
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Unformatted text preview: D ( n,m ) = D ( n-1 ,m ) + D ( n,m-1) (5) How many permutations are there of a poker deck, in which the second ace (that is, second in the order of the permutation) is at the k th position. ±or what k is this number the greatest? (I.e. if you want to bet that from a shu²ed poker deck, what is the position of the second ace, what would be the bet that maximizes the probability of winning?) 1...
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