# Counting the number of derangements

im missing something and i cant figure out what. Ive seen a proof of the formula for the number of derangements (permutations with no fixed points) using inclusion-exclusion and it makes perfect sense to me. I just cant wrap my head around why it isn't much simpler than that.

By my logic, assuming F is the permutation, the first element of {1,...,n} has n-1 possible mappings, the second element can be mapped anywhere except for itself and F(1), thus n-2 possibilities and so on. So why isn't the number of derangements simply (n-1)!?

"the second element can be mapped anywhere except for itself and F(1)". sure, but what if the second element happens to be $F(1)$? You could rule this out, but then your "and so on" clause needs a little rephrasing
What if $F(1) = 2$?