I want to first thank the people who have been patiently helping me with proofs. Your kind help has paid off and the lecture I attended tonight really clarified things.

Now for the new problem:

How many ways are there to seat five women and five men around a circular table if the seating alternatives man-woman-man-woman, etc.?

First pair everyone as couples

and now seat the couples around the table. There are 5 couples so that makes 5!/5 ways to seat the couples around the table. Now we need to rearrange the men or rearrange the women while leaving the other sex where they are. I calculate 5! ways to permute the couples. My answer would be 5!*5!/5.

Have I made a mistake?