A group consists of 4 couples in which each of the 4 persons have one wife each. In how many ways could they be arranged in a straight line such that the men and women occupy alternate positions?

For the 4 men and 4 women to sit in alternate positions, the row must either start with a man or start with a woman. If the row starts with a man, the 4 men can be arranged in the 4 odd positions in 4 factorial ways, and the 4 women can be arranged in the 4 even positions in 4 factorial ways, giving 4 factorial times 4 factorial arrangements. Because either gender can start the row, we multiply this by 2, resulting in 2 times 24 times 24, which is 1152. The result is 1152.

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