SOLUTION: ten people, 5 men and 5 women, are to be seated in a row of 10 chairs. What is the probability that the men and women end up in alternate chairs?

First we find the numerator of the probability.

The number of ways we can have this arrangement

M W M W M W M W M W

For each of the 5! ways to place the men there are 5! ways to place the
women. That's 5!*5! or (5!)²

But we could also have this arrangement:

W M W M W M W M W M

which is another (5!)²

So the numerator, which is the number of ways to have men and women in
alternate seats, is 2(5!)²

The denominator is the number of ways anybody can sit anywhere, which
is 10!

So the probability is


                           


Edwin