Four men and four women are forming a circle for a folk dance. In how many ways can this be done if we require that the men and women alternate positions?
My attempt was to square $24$ for total possibilities $= 576$. Then I tried to find out how many possibilities to subtract to find the answer. I got nowhere. I did find a couple of equations concerning circles and rings but they seemed contradictory:
- The number of ways $n$ distinct objects can be arranged in a circle is $(n-1)!$.
- The number of ways $n$ distinct beads (with $n\geq 3$) can be placed on a necklace is $(n-1)!/2$.
I don't see the difference.
Thank you for any insight you can give me.