/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Given \(n\) distinct objects, de... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 24

Given \(n\) distinct objects, determine in how many ways \(r\) of these objects can be arranged in a circle, where arrangements are considered the same if one can be obtained from the other by rotation.

Short Answer

Expert verified
The number of arrangements of \(r\) objects from \(n\) distinct ones in a circular arrangement is given by the formula \(\frac{(n-1)!}{(n-r)!}\).

Step by step solution

01

Understand Problem Statement

The task asks to find the number of arrangements of \(r\) objects out of \(n\) distinct objects, when arranged in a circle. Here, rearrangements by rotation are considered the same. Thus, we are dealing with a problem of circular permutations.
02

Apply Circular Permutation Formula

The formula for circular permutations of \(r\) objects from \(n\) distinct objects is \((n-1)P(r-1)\) or equivalently \(\frac{(n-1)!}{(n-r)!}\). This formula takes into account that the circular arrangement has rotational symmetry.
03

Generalize the Solution with Variables

In order to give a generalized solution to the problem, it remains to write down the formula for circular permutations using the variables \(n\) (number of distinct objects) and \(r\) (number of chosen arrangements) as follows: The number of circular permutations = \(\frac{(n-1)!}{(n-r)!}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Four numbers are selected from the following list of numbers: \(-5,-4,-3,-2,-1,1,2,3,4\). (a) In how many ways can the selections be made so that the product of the four numbers is positive and (i) the numbers are distinct? (ii) each number may be selected as many as four times? (iii) each number may be selected at most three times? (b) Answer part (a) with the product of the four numbers negative.

Determine the value of each of the following summations. a) \(\sum_{i=1}^{6}\left(i^{2}+1\right)\) b) \(\sum_{j=-2}^{2}\left(j^{3}-1\right)\) c) \(\sum_{i=0}^{10}\left[1+(-1)^{i}\right]\) d) \(\sum_{j=0}^{4}\left(3^{\prime}-2^{\prime}\right)\) e) \(\sum_{k=2}^{4}(-1)^{k}\) f) \(\sum_{k=n}^{2 n}(-1)^{k}\), where \(n\) is an odd positive integer g) \(\sum_{i=1}^{6} i(-1)^{i}\)

a) In how many ways can the letters in UNUSUAL be arranged? b) For the arrangements in part (a), how many have all three U's together? c) How many of the arrangements in part (a) have no consecutive U's?

a) In how many ways can we select five coins from a collection of 10 consisting of one penny, one nickel, one dime, one quarter, one half-dollar, and five (identical) Susan B. Anthony dollars? b) In how many ways can we select \(n\) objects from a collection of size \(2 n\) that consists of \(n\) distinct and \(n\) identical objects?

a) In how many ways can seven people be arranged about a circular table? b) If two of the people insist on sitting next to each other, how many arrangements are possible?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.