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A group of $n$ couples (a total of $2n$ people) sit at a circular table. Arrangements that differ by any rotation of the seating positions are considered to be the same. Find a formula for the number of seatings where no woman sits beside her partner?

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  • $\begingroup$ By complement, we find the number of seating where there are one couples sits together. We choose one couples from n couple and fixed them at two seats. So, the number to arrange is 2(2n-2)!. So, we can find the number of seating where no woman sits beside her partner are (2n-1)!-2(2n-2)!= (2n-3)*(2n-2)!. $\endgroup$ – Worawit Tepsan Oct 31 '13 at 6:31
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This can be solved using inclusion-exclusion. We have $n$ conditions for the $n$ couples sitting next to each other. If one of the conditions is fulfilled, we can regard the corresponding couple as a single person to be seated and include a factor $2$ for the two orders in which the couple can sit. So there are $2^k(2n-k-1)!$ different arrangements that satisfy $k$ particular conditions, and thus

$$ \sum_{k=0}^n\binom nk(-1)^k2^k(2n-k-1)! $$

different arrangements that satisfy none of the conditions. This is OEIS sequence A129348.

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