# Arrangement of couples

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• Oct 17th 2009, 01:52 AM
DeRSeD
Arrangement of couples
How many ways are there to seat n married couples in a row of 2n chairs such that the couples never sit next to each other?

Shouldnt it be as as simple as n.C.2n divided by 2!
• Oct 17th 2009, 05:17 AM
Plato
Using inclusion/exclusion $\sum\limits_{k = 0}^n {\left( { - 1} \right)^k \binom{n}{k}\left( {2n - k} \right)!\left( {2^k } \right)}$.
• Oct 17th 2009, 06:42 AM
aman_cc
Quote:

Originally Posted by Plato
Using inclusion/exclusion $\sum\limits_{k = 0}^n {\left( { - 1} \right)^k \binom{n}{k}\left( {2n - k} \right)!\left( {2^k } \right)}$.

@ Plato - Hi

Did you actually workout all the cases - "1,2,3,...,n couple together" and took a compliment or there was an easy way?

Thanks
• Oct 17th 2009, 06:58 AM
Plato
Quote:

Originally Posted by aman_cc
Did you actually workout all the cases - "1,2,3,...,n couple together" and took a compliment

Exactly. I do not know another way.
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