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 $i\neq j$?