PS02-solutions

PS02-solutions - MATH 681 Problem Set #2 Solutions 1. (10...

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MATH 681 Problem Set #2 Solutions 1. (10 points) Prove the following identity combinatorially: n X i =0 i ± n i ² = n 2 n - 1 Both sides can easily be interpreted as enumerating the ways to select an element x of { 1 , 2 , 3 ,...,n } , and then choose a subset S from among the remaining n - 1 elements. The right side of this equation clearly enumerates the set of ways to do the above: there are n choices for x , and, regardless of the choice of x , S is a subset of a set of size n - 1; thus there are 2 n - 1 choices of S . To calculate the left side of the equation, we begin by choosing a set T which will contain both S and x , then we select x as an element of T and deﬁne S = T - { x } . Thus, one may uniquely determine x and S by, instead of choosing x then a set S not containing x , choosing a set T and an element thereof to serve as x . For any value of i , there are ( n i ) possible choices of T of size i ; then, having schosen a set T of size i , any of its i elements can serve as a value of x , so there are i ( n i ) selections of T and x given | T | = i . Ranging over all possible sizes of T , we see that there are n

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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.

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PS02-solutions - MATH 681 Problem Set #2 Solutions 1. (10...

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