I have an 8x8 chess board and place my 32 pieces randomly on the board (ignoring any chess rules). What is the probability that every piece is on a square that matches its colour?
My attempt:
I have to choose 32 squares on which to place my pieces. This can be done in ${64 \choose 32}$ ways. But for each set I choose I can place my pieces in $32!$ ways so the total number of ways to place my pieces is $${64 \choose 32}32!$$
I need the number of white squares I choose to equal the number of black squares (i.e. 16 of each). How many ways can I choose that satisfy this requirement? I don't think I care about order so I'll choose my 16 black squares first and then my 16 white squares. I can choose the first black square in 32 ways, the second in 31 ways ... the 16th in 17 ways, then I do the same for the white squares. So the total number of ways with equal numbers of black and white squares is $$\left(\frac{32!}{16!}\right)^2$$.
Assuming all black pieces are identical (likewise for white) there is only one way to place my pieces so that the colours match. Therefore the overall probability is $$\frac{\left(\frac{32!}{16!}\right)^2}{{64 \choose 32}32!}$$
I'm not confident if this is right because I get hopelessly confused in combinatorics with when I should assumed things are identical, and whether I've double counted something.