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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 s 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 inFnite repetition number. Prove that
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.

Ask a homework question - tutors are online