Possible Duplicate:
Circular Permutation
I was looking through old homework to work on my counting skills, and I had this problem that I didn't get right.
Question: In how many ways can 4 men and 5 women be seated around a circular table so that no 2 men are seated next to one another?
My Solution: First we sit the women, and leave an open spot in between each to assure a man doesn’t sit next to a man. There are 5! ways to do this. The there are 5 open spots, we only want to choose 4 of those for men to stay in, so $5 \choose 4$. Then there are 4! possible different ways to arrange the men 4 within those spaces. Finally, since the table is round, we want to make sure that rotating it doesn’t make a difference. We can rotate the table 9 times that will produce the same arrangement, thus divide by 9. With the product rule, we find that the total number of arrangements to be:
$\frac{5!{5 \choose 4}4!}{9} = 1600$
I believe I undercounted the total arrangements for this problem, but I am not sure where I went wrong.