# How many ways are there to order $n$ women and $n$ men in circle

I have the following question :

How many ways are there to order $n$ women and $n$ men in circle so there is no man next to man and no woman next to man meaning the order is man,woman,man,woman...

This is how I started :

Lets sit down all the women in circle therefore we get $(n-1)!$ now we have $n$ men we know that each woman could have $n$ different men sitting next to her on one side and $n-1$ men at the other side and we know that sitting $n$ men in circle is $(n-1)!$, now I don't know to create a "law" to make the order man,woman,man,woman...

Any ideas how to approach this?

Thanks

## 2 Answers

Suppose one of the women is named “W” and one of the men is named “M”. Now you have already explained that there are $(n-1)!$ ways to arrange the women in a circle, and similarly $(n-1)!$ for the men. Any circular arrangement of men can be “interleaved” with any circular arrangement of women in $n$ different ways (“M” can be the first, second, third, ..., or $n$th man to the right of “W”, so there are $n\cdot(n-1)!\cdot(n-1)!$ arrangements.) These are all different arrangements, because either the circle of men is different, the circle of women is different, or the number of places between “W” and “M” are different.

• Thank you for your answer, just to making sure I got it correctly you placed all the women and men to sit meaning $(n-1)!*(n-1)!$ since each women as well as $"W"$ has $n$ different men that could sit near to her (or we could say that each men has $n$ different women that could sit near to him) then we multiply by $n$? – JaVaPG May 22 '16 at 21:03
• Yes, that is basically it. Once you know the circular orders separately of the men and of the women, you only need to specify one more thing: which of the $n$ men is at the immediate right of woman “W” to fully specify the arrangement. – Steve Kass May 22 '16 at 21:09

There are two scenarios here. Either the first person is a man or the first person is a woman. In each of the scenarios you now have to order two groups of $n$ objects into $n$ places. That means there are $n!$ different combinations and $n!$ for women. So there are $2(n!)^2$ ways to order them by those rules.

• I think you have overlooked the fact that the people are in a circle, which presumably means that two configurations that differ only by a rotation are considered equivalent. – Steve Kass May 22 '16 at 20:45