# Combinatorics- dividing animals

We have n pairs of different animals (male and female from each species, so each pair is different). How many ways are to divide the $$2n$$ animals into $$n$$ different rooms, so in every room there are exactly two animals that are not from the same species? (two females and two males are allowed).

I know I should use the Inclusion exclusion principle but I don't know how... Thank you:)

Hint: Call $$A_i=\{\text{Ways to pair them such that animals i got the same room}\}$$ Then you want all possible pairs minus the union of these $$A_i$$ i.e., $$|A\setminus \bigcup _{i=1}^n A_i|.$$ To count all possible pairings do all permutations and split them 2 by 2 but divide by $$2^n$$ because there is no order inside the rooms.
Can you count $$A_i$$? Choose a room for the $$i-$$th animals and scratch that room and the animals. Do the same to count all pairings but now you have $$n-1$$ pairs and $$n-1$$ rooms. What about $$A_i\cap A_j$$ for $$i\neq j$$?