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

Question

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.

Zero? The tenth letter should go into the correct envelope as well. — Srivatsan, Aug 30 '11 at 13:59
possible duplicate of How many fixed points in a permutation — Aryabhata, Aug 30 '11 at 14:05
@Tretwick , André probably wants you to think about what exactly the question is asking more deeply. If 9 letters are in the correct envelope, how many letters are in incorrect envelopes? — Ragib Zaman, Aug 30 '11 at 14:05
Related: http://math.stackexchange.com/questions/17320/derivation-of-the-partial-derangement-rencontres-numbers-formula — Aryabhata, Aug 30 '11 at 14:05
@Aryabhata I thought so too, but this question is more like a puzzle :-) — Srivatsan, Aug 30 '11 at 14:05
@Srivatsan: I was going by the title and was addressing the general question. — Aryabhata, Aug 30 '11 at 14:07
It has already been solved by @Srivatsan Narayanan. Can exactly $9$ be in the right envelope? No, because there is only one letter left over, and the right envelope for that letter. — André Nicolas, Aug 30 '11 at 14:07
Something less abstract than envelopes and letters. Ten married couples. All $20$ people are dancing, in pairs. Could exactly $9$ women be dancing with their husbands? Who is left over for the $10$th woman to dance with? — André Nicolas, Aug 30 '11 at 14:12

Srivatsan · Accepted Answer · 2011-08-30T14:19:25.107

5

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.

edited Aug 30 '11 at 14:19

answered Aug 30 '11 at 14:04

Srivatsan

26,311

1

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

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

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