# Problem about $n$ couples sitting at a round table

In how many ways can $n$ couples sit at a round table such that no couple is sitting opposite each other?

I know that if we were to only arrange $n$ couples around a table we get $(2n-1)!$. But I don't know how to do this question.

• So you want $2n$ people in a circle such that no two people from the same couple are sitting opposite each other? What exactly is the $(n-1)!$ counting? – TMM Apr 27 '13 at 14:43
• (n-1)! is counting if we arrange the couples around the circular table but allowing couples to sit opposite to each other. – please delete me Apr 27 '13 at 15:37
• How is that $(n - 1)!$? There are $2n$ people, so I suppose the answer would be closer to $2n!$. Or should couples sit on the same seat? – TMM Apr 27 '13 at 16:11
• it would be $(2n-1)!$ because it's a circular table. – pad Apr 27 '13 at 16:20
• Sorry, yes I meant $$(2n-1)!$$ But this isn't the solution to this question. How do I do this question? – please delete me Apr 27 '13 at 16:51

If we have n couples, we have 2n people. And we can arrange 2n people around a circular table in $\frac{2n!}{2n}$ = $(2n-1)!$ For n-1 couples say we have $a_{n-1}$ arrangements. When we go to the n case we can put the first spouse in $2n-2$ places and then we can put the second in $2n -2$ places also. so we get $a_n = a_{n-1}(2n-2)^2$ Rolling back we get $a_n = a_1(2n-2)^2(2n-4)^2...(2)^2 =4^{n-1}(n-1)^2(n-2)^2...(1)^2 =4^{n-1}(n-1!)^2$
• @Alexander: This isn’t right, I’m afraid. Suppose that there are $3$ couples, $aA,bB$, and $cC$. If $a$ and $A$ sit opposite each other, there are $4!$ ways to seat the remaining two couples relative to $a$ and $A$. Similarly, there are $4!$ seatings with $b$ and $B$ opposite each other, and $4!$ with $c$ and $C$ opposite each other. If two couples are seated opposite each other, so is the third, and there are $2^3$ such seatings, each of which has been counted $3$ times, so the net number of unacceptable seatings is $3\cdot4!-2\cdot2^3=56$, not $4!$. – Brian M. Scott Apr 27 '13 at 18:38