I have the following question :
How many ways are there to order in circle $n$ couples so each men sits infront of his wife?
I thought of something like that :
Lets take wife $n$ and sit her down so the rest will be ordered according to her. now we have $n-1$ wifes and $n$ men, lets sit down all the wifes the left $n-1$ since we have a circle of $2n$ people ($n$ men, $n$ women) we need to sit down the rest of wifes in $2n-1$ sits (since the first women already sits) meaning choose $n-1$ wifes to sit in $2n-1$ sits, the rest of the husbands has one place to sit (infront of his wife)
I think my method is correct but when I choose $n-1$ from $2n-1$ could I get husband and wife instand of just the wifes? I mean I don't understand how choosing $n-1$ from $2n-1$ creates that I'll only choose wifes since if I also choose husbands I might take the husband with his wife and then I don't get that the rest has one place to sit.
Any ideas? Thanks!