# Thread: Ways to seat men and women alternately around a circular table

1. ## Ways to seat men and women alternately around a circular table

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 $\displaystyle (M_{1}W_{1})(M_{2}W_{2})(M_{3}W_{3})(M_{4}W_{4})(M _{5}W_{5})$ 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.

2. Originally Posted by oldguynewstudent
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 $\displaystyle (M_{1}W_{1})(M_{2}W_{2})(M_{3}W_{3})(M_{4}W_{4})(M _{5}W_{5})$ 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.

Hmm let me try another way and see if the answer comes out the same.

Fix the position of one of the men, then there are 4! ways to arrange the other men. Now consider the position to the left of the first man, and place a woman there; there are 5 ways to do this. Then there are 4! ways to arrange the other women. So I get 4!*5*4!, which is the same as your answer.

I admit I have a hard time following your method, but it seems right.