# In how many ways can $10$ letters be placed in $10$ addressed envelope such that exactly $9$ letters are in correct envelope?

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.

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Zero? The tenth letter should go into the correct envelope as well. –  Srivatsan Aug 30 '11 at 13:59
I don't have the answer/solution for this one. –  Quixotic Aug 30 '11 at 14:01
There is less to this problem than meets the eye. –  André Nicolas Aug 30 '11 at 14:02
@André Nicolas:Pardon,what exactly do you mean? –  Quixotic Aug 30 '11 at 14:03
possible duplicate of How many fixed points in a permutation –  Aryabhata Aug 30 '11 at 14:05

## 1 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.

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It has to go the correct envelope. –  Quixotic Aug 30 '11 at 14:06
Yes. So, the answer is... –  Srivatsan Aug 30 '11 at 14:07
$0$,thanks a lot for your inputs :-) –  Quixotic Aug 30 '11 at 14:15