In how many ways can four men and four women form a circle if the men and women alternate positions? 
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.
T. Grode
 A: Hint: Apply multiplication principle for the following steps:


*

*Arrange the men in a circle.  Have the men leave a bit of extra space inbetween themselves.

*Once the previous step is completed, place a woman into each empty space.

 The first step can be completed in $3!$ ways.  The second step can be completed in $4!$ ways.  Applying multiplication principle, there are then $3!\cdot 4!$ arrangements.

We can generalize this to $n$ men and $n$ women.

 There will be $(n-1)!$ ways to do the first step.  There will be $n!$ ways to do the second step.  There will then be $(n-1)!\cdot n!$ total number of arrangements.


In counting how many options there are to complete the first step, we may recognize that we may divide by symmetry or we can come up with a more clever way of describing how to accomplish the first step without overcounting.  For example:


*

*Set the man whose name appears first in alphabetical order on the dancefloor, it matters not where.

*Pick one of the remaining men to stand in clockwise position to the first man.

*Pick another of the remaining men to stand in the next available position clockwise.

*Repeat this process until all men are placed.

