/*! 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 19 In how many ways can three men a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 19

In how many ways can three men and three women be seated around a circular table (that seats six) assuming that women and men must alternate seats?

Short Answer

Expert verified
12

Step by step solution

01

Fix a Reference Point

In circular permutations, one position needs to be fixed to avoid counting rotations as different arrangements. Fix the position of one of the men.
02

Arrange the Remaining People

With one man fixed, there are two remaining men and three women to be arranged in the alternating seats pattern. Consider the seats to alternate: M-W-M-W-M-W, where M is a man and W is a woman.
03

Permute the Remaining Men

There are 2 remaining men to place in the M slots. The number of ways to arrange these two men is given by: \[ P_2 = 2! = 2 \]
04

Permute the Women

There are 3 women to place in the W slots. The number of ways to arrange these three women is given by: \[ P_3 = 3! = 6 \]
05

Combine the Arrangements

The total number of ways to arrange the men and women around the table is the product of the permutations of men and women: \[ 2 \times 6 = 12 \]

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a branch of mathematics that studies counting, arrangement, and combination of objects. In this problem, we use combinatorics to determine the number of ways to arrange three men and three women around a circular table. This involves considering different permutations and combinations while ensuring that men and women alternate in the seating arrangement. By breaking down the problem into simpler steps and using combinatorial principles, we can systematically count the valid seating arrangements.
Factorial
Understanding factorials is crucial in solving this problem. A factorial, denoted as \(!n\), represents the product of all positive integers up to \(!n\). For example: \(2! = 2 \times 1 = 2\) and \(3! = 3 \times 2 \times 1 = 6\).

In this exercise, the factorials help us count the permutations of men and women. By fixing one man to avoid redundancy from rotations, we consider the remaining people. The number of ways to arrange the remaining men is \(!2 = 2!\) and for the remaining women is \(!3 = 3!\). By multiplying these permutations, we get the total number of valid seating arrangements.
Alternating Arrangement
In this problem, we need an alternating arrangement where men and women sit in alternating seats. This ensures that no two men or two women are seated next to each other. To achieve this:
  • Fix one man in a seat to set a reference point.
  • Arrange the remaining men in the available man slots.
  • Arrange the women in the remaining slots.
Since the men and women need to alternate, the pattern will be M-W-M-W-M-W. By following these steps and combining the individual permutations of men and women, we arrive at the total number of ways to achieve this alternating seating arrangement around the circular table.

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

Suppose that a local area network requires eight letters for user names. Lower- and uppercase letters are considered the same. How many user names are possible for the local area network?

How many different license plate numbers can be made by using one letter followed by five digits selected from the digits 0 through 9?

Suppose your financial advisor recommends three stocks to you. He claims the likelihood that the first stock will increase in value at least \(10 \%\) within the next year is \(0.7,\) the likelihood the second stock will increase in value at least \(10 \%\) within the next year is \(0.55,\) and the likelihood the third stock will increase at least \(10 \%\) within the next year is \(0.20 .\) Would it be unusual for all three stocks to increase at least \(10 \%,\) assuming the stocks behave independently of each other?

Four members from a 20-person committee are to be selected randomly to serve as chairperson, vice-chairperson, secretary, and treasurer. The first person selected is the chairperson; the second, the vice-chairperson; the third, the secretary; and the fourth, the treasurer. How many different leadership structures are possible?

According to the Sefton Council Password Policy (August 2007), the United Kingdom government recommends the use of 鈥淓nviron passwords with the following format: consonant, vowel, consonant, consonant, vowel, consonant, number, number (for example, pinray45).鈥� (a) Assuming passwords are not case sensitive, how many such passwords are possible (assume that there are 5 vowels and 21 consonants)? (b) How many passwords are possible if they are case sensitive?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Statistics

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.