Question

In how many ways can 5 men and 5 women sit around the table if there is no two women are adjacent?

Answer 1

NOTE: The following assumes that a unique state is defined by the chair a person is sitting at. You used the word "around" which hints that only relative position matters, which would change the answer.


Really messy and I apologize, but think of it like a row of chairs. In the first chair, anyone can sit down. Total number of possibilities is 10. In the next chair, only members of the opposite sex can sit down, of which there are five. 5*10 possibilities. Same constraints, only this time there are 4 possibilities. Why? In the very first round, someone from this gender was already used. This pattern continues.

10 * 5 * 4 * 4 * 3 * 3 * 2 * 2 * 1 * 1
Notice how the last two chairs are forced, which makes sense (think about what it would mean if it weren't forced). Evaluating this, we get: 28 800.