Divorced Couples Problems Combinatorics

Question

5 couples are sitting in a line but no one wants to sit next to their partner. How many ways can this be done?

Here's my solution. Is this correct? $$10!-{5 \choose 1}2! \cdot 9!+{5 \choose 2}2!^2 \cdot 8!-{5 \choose 3}2!^3 \cdot 7!+{5 \choose 4}2!^4 \cdot 6!-{5 \choose 5}2!^5 \cdot 5!$$

Answer 1

Yes, you're converting the current number of couples under consideration into an ordered element and then ordering the elements. Looks fine.