Assume that four men and four women are to sit in a row of eight chairs. In how many ways can these people sit if all men must sit together and all the women must sit together? Use the fundamental counting principle. To apply this principle, first identify the separate tasks involved in making the seating arrangements.

The people can sit in 1,152 possible ways if all men must sit together and all women must sit together.

Explanation:

The 4 men must sit together.

Number of ways to arrange the 4 men together

= 4! = 4 x 3 x 2 x 1 = 24

Similarly, the 4 women must sit together.

Number of ways to arrange the 4 women together

= 4! = 4 x 3 x 2 x 1 = 24

Now the 8 chairs are placed in a row. There are 2 ways to arrange the men and women: either the men must sit in the first 4 chairs and women in the last 4 chairs, or the women must sit in the first 4 chairs and men in the last 4 chairs.

Hence, total number of ways to arrange the 8 people

= 24 x 24 x 2

= 1,152