Edit: You may ignore this incorrect solution, I leave it for people to see where I did wrong.

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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)!