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.

  • 7
    $\begingroup$ Zero? The tenth letter should go into the correct envelope as well. $\endgroup$
    – Srivatsan
    Aug 30, 2011 at 13:59
  • $\begingroup$ I don't have the answer/solution for this one. $\endgroup$
    – Quixotic
    Aug 30, 2011 at 14:01
  • 4
    $\begingroup$ There is less to this problem than meets the eye. $\endgroup$ Aug 30, 2011 at 14:02
  • $\begingroup$ @André Nicolas:Pardon,what exactly do you mean? $\endgroup$
    – Quixotic
    Aug 30, 2011 at 14:03
  • $\begingroup$ possible duplicate of How many fixed points in a permutation $\endgroup$
    – Aryabhata
    Aug 30, 2011 at 14:05

1 Answer 1


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.

  • 1
    $\begingroup$ It has to go the correct envelope. $\endgroup$
    – Quixotic
    Aug 30, 2011 at 14:06
  • $\begingroup$ Yes. So, the answer is... $\endgroup$
    – Srivatsan
    Aug 30, 2011 at 14:07
  • $\begingroup$ $0$,thanks a lot for your inputs :-) $\endgroup$
    – Quixotic
    Aug 30, 2011 at 14:15

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