I have one problem which goes like this: "In how many ways can $10$ letters be placed in $10$ addressed envelope such that exactly $9$ letters are in correct envelope?"
If I understand the problem correctly this is similar to counting derangement with exactly $r$ matches,I don't know how to do it,please help.
Best Answer
If nine letters go into the correct envelopes, what can you say about the remaining 1 letter?
The following is not necessary for solving this problem, but I am adding it since you mentioned derangements and number of permutations with exactly $k$ matches (aka fixed points). The more general problem is to find the number of permutations with exactly $k$ fixed points. The solution for this is described in this wikipedia page.