How many ways are there to seat n married couples at a round table with 2n chairs in such a way that the couples never sit next to each other?
Edit: You may ignore this incorrect solution, I leave it for people to see where I did wrong.
Firstly, we will find the number of the ways that the couples sit together.
As there are n couples, there will be (n-1)! different arrangements for the couples. And each couple can sit in 2! ways.
If a couple can sit in 2 ways, n couples can sit in 2^n ways.
Hence, n couples can be placed in 2^n (n-1)! ways. (when every couple sits together)
All arrangements - Couples sitting together = Couples not sitting together
We can place 2n people in (2n-1)! ways.
So, the answer is
All arrangements - Couples sitting together
(2n-1)! - 2^n (n-1)!