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Chapter 33: Problem 1072

In how many ways may a party of four women and four men be seated at a round table if the women and men are to occupy alternate seats?

Short Answer

Expert verified
The total number of arrangements for a party of four women and four men to be seated at a round table if the women and men are to occupy alternate seats is 144 ways.

Step by step solution

01

Fix one person's position

In a circular arrangement, fix one person's position (say a woman) to avoid duplicate counting due to rotation. Now, there are 3 remaining seats for the other women.
02

Arrange the women

There are 3 women left to be seated and 3 seats available for them. The number of ways to arrange them is the number of permutations of these 3 women, which is 3!.
03

Arrange the men

Since the men and women are to occupy alternate seats, there are now 4 seats left for the men. The number of ways to arrange the 4 men is the number of permutations of these 4 men, which is 4!.
04

Calculate the total number of arrangements

Since the arrangements of men and women are independent of each other, we can multiply the number of ways to seat the women and the number of ways to seat the men to obtain the total number of arrangements. Total number of arrangements = (Number of ways to seat women) * (Number of ways to seat men) Total number of arrangements = \(3! * 4!\) Now, let us calculate the values for 3! and 4! and find the total number of arrangements.
05

Calculate the factorial values

\(3! = 3 * 2 * 1 = 6\) \(4! = 4 * 3 * 2 * 1 = 24\)
06

Final Answer

Now, we can find the total number of arrangements by multiplying the factorial values calculated above. Total number of arrangements = \(3! * 4! = 6 * 24 = 144\) So there are 144 ways for a party of four women and four men to be seated at a round table if the women and men are to occupy alternate seats.

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