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Expert: - 11/28/2011


Question
How do you solve this question? i got the answer for it on the link but i'm not sure how you get there.
http://img690.imageshack.us/img690/7463/answerlb.png

two people of the same sex could sit side-by-side.
If in addition to the condition that no husband and wife are
side-by-side we require that no two people of the same sex are
side-by-side, we obtain a famous problem known as the menage
problem. Solve this problem.

Answer
I don't find that address.

I will assume we have n couples.
The number of ways to seat the men is n!.
The number of ways to seat the 1st woman is n-2.
One or two couples could not be done.

Lets take 3 couples.  There would be 2 ways to seat the men, since they have to sit every other chair and could be seated clockwise or counterclockwise.  Once they have been sat down, there is only 1 way to sit the women, so the total ways is 2.

If there are 4 couples, the men give 3! = 6 combinations for the men.
Given where they are, there are only two ways to put in the girls, so 2*6=12 total ways.

If there are 5 couples, the men have 4! = 24 ways to sit.
For each of these, I find 12 ways to sit the women, and 12 = 2*3!.


If there are n couples, the men have (n-1)! ways to sit.
For each of these, I find 2(n-2)! ways for the women to sit.
This gives a total of 2(n-1)!(n-2)!

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