Arrangement of couples

**DeRSeD** · October 17th 2009, 12:52 AM

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!

**Plato** · October 17th 2009, 04:17 AM

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

**aman_cc** · October 17th 2009, 05:42 AM

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

**Plato** · October 17th 2009, 05:58 AM

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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