Hi guys ,
I have some combinatoric problems. There are 6 couples. All the 6 wives are to sit together. Find the number of ways if the none of the wives are sitting next to her husband? Thank you very much.
Regards
Pang
What does that mean? Is it a circular table? Are they sitting in a row? Are they just sitting randomly (wives in a gaggle, husbands spread out throughout the room randomly and just plop down when they feel like it)? The answer depends on how they are seated.
So, there are 12 seats in a row. You have six women sitting in a group. We have the following arrangements:
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The first and last arrangement are similar. The middle 4 are similar. For the first and last case, we permute the wives. The only open seat that is next to a wife can go to any husband but her own. Then we permute the remaining husbands:
$2\cdot 6!\cdot 5\cdot 5!$
For the rest of the cases, there are two possibilities. The husband of the wife on the right is seated next to the wife on the left or he isn't. So,we permute the wives, there are 4 husbands that are not married to either wife "on the end". Then, then, there are 4 husbands not married to the wife on the right, then we permute the remaining 4 husbands. Or, the husband of the woman on the right sits next to the woman on the left. Then there we can simply permute the remaining 5 husbands. This gives:
$5\left( 6!\cdot 4\cdot 4\cdot 4!+6!\cdot 5!\right) $
Total:
2,678,400