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CSCE 222 CSCE222 Problem Set 8 – Texas A&M

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CSCE 222 CSCE222 Problem Set 8 – Texas A&M

Resources.(All people, books, articles, web pages, etc. that have been con-sulted when producing your answers to this homework)

Discrete Mathematics and Its Applications Seventh Edition by Kenneth H.Rosen

In this problem set, you will earn total 100 + 20 (extra credit) points.

Problem 1.Section 6.4, Exercise 38, page 422

Solution.

Each term on the lefthand side of the equation counts the numberof ways to choose a subsetSofkelements of{1,2, …, n}and then a sequenceof two not necessarily distinct elements fromS.

This means that it counts thenumber of ways to select a triple (S, x, y) whereSis a subset of{1,2, …, n}andxandyare not necessarily distinct elements fromS.

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CSCE 222 CSCE222 Problem Set 8 – Texas A&M

Resources.(All people, books, articles, web pages, etc. that have been con-sulted when producing your answers to this homework)

Discrete Mathematics and Its Applications Seventh Edition by Kenneth H.Rosen

In this problem set, you will earn total 100 + 20 (extra credit) points.

Problem 1.Section 6.4, Exercise 38, page 422

Solution.

Each term on the lefthand side of the equation counts the numberof ways to choose a subsetSofkelements of{1,2, …, n}and then a sequenceof two not necessarily distinct elements fromS.

This means that it counts thenumber of ways to select a triple (S, x, y) whereSis a subset of{1,2, …, n}andxandyare not necessarily distinct elements fromS.

When k elements are selected, there will be(nk)possibilities for the selection of subset S, and therewill be k2 possibilities for the selections o fxandy.The summation of anyinteger forkfrom 0 tongives the number of ways to select this triple (S, x, y).

The righthand side of the equation is expressed asn(n+ 1)2n2which is equalton(n2n2+ 2n2), which is equal ton(n2n2+ 2n12n2) since 2n2is thesame thing as 2n1/2, which means 2n12n2= 2n2.

This expression isequal ton((n1)(2n2) + 2n1) which isn(n1)2n2+n2n1. There are twocases to show the righthand side.

Case 1: Pick the same element from the subset twicePicking that element from the set ofnelements means there is a total ofnpossibilities.

Picking the rest of the subset, there are a total of 2n1possibilitiesto pick a given subset since there aren1 elements left. By product rule, the total possibilities is the product of the two. Thus, the total possibilities is n2n1

CSCE 222 CSCE222 Problem Set 8 – Texas A&M

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