I'm trying to solve the misaddressed letters problem (number of permutations on $\{1,...,n\}$ such that $k$ is not in the $k$-th place) in the following way and something should be wrong: total number of permutations: $n!$ now I want to substract:
permutations with exactly one correct place: $n (n-2)!$
permutations with exactly two correct places: ${n \choose 2} (n-3)!$
... permutations with exactly s correct places: ${n \choose s} (n-(s+1))!$
What am I missing? thanks