Question

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

There are 144 ways for the party of four women and four men to be seated at the round table with alternate seating arrangements.

Step-by-step solution

1. Arrange the Women

Since the seats are arranged in a circle, there are different ways to count the arrangements. One way is to fix one woman's position and then arrange the other three women around her. There are then 3! ways to arrange the remaining women (3 possible seats for the first woman, 2 for the second, and 1 for the last). Thus, there are 3! = 6 ways to place the women.

2. Arrange the Men

Once the women are seated, the men must occupy the remaining 4 alternate seats in between the women. We treat these seats as a fixed set of positions. So there are 4 positions to place the first man, 3 for the second, 2 for the third, and 1 for the last. Therefore, there are 4! = 24 ways to place the men.

3. Calculate the Total Number of Arrangements

To find the total number of seating arrangements, we simply multiply the number of ways to arrange the women (6) by the number of ways to arrange the men (24). So, there are 6 * 24 = 144 ways for the party of four women and four men to be seated at the round table with alternate seating arrangements.