   Chapter 9.3, Problem 43ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# A derangement of the set {1,2,….,n} is a permutation that moves every element of the setaway from of {1,2}, and 231 and 312 are derangements of {1,2,3}. For each positive integers n, let d n be the number of derangements of the set {1,2,….,n}. (a) Find d 1 , d 2 and d 3 . (b) Find d 4 Find a recurrence relation for d 1 , d 2 , d 3 , ...

To determine

(a)

To find the value of d1, d2 and d3.

Explanation

Given information:

A derangement on the set {1,2,....,n} is a permutation that moves every element on the set away from its “natural” position. Thus 21 is a derangement of {1,2} and 231 and 312 are derangements of the set {1,2,3}. For each positive integer n, let dn be the number of derangements of the set {1,2,...,n}

Calculation:

For d1, the value of n must be equal to 1. The set with only one element is {1}. The permutation of this set is only one because there is no other permutation possible with only one element. Therefore, there is no derangement possible for one element. Thus value of d1=0.

For d2, the value of n must be equal to 2

To determine

(b)

To find the value of d4

To determine

(c)

To find the recurrence relation for d1,d2,d3,......

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