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Possible Duplicate:
Number of permutations where n ≠ position n

In how many ways can 5 messages be posted in 5 envelopes so that no correct message is posted, if each envelope only has one correct message from the 5, and each message only has one correct envelope?

Possible notation: We represent each envelope by an upper-case letter, $A, B, C, D, E$, and each message by a lower-case letter, $a,b,c,d,e$, and denote the positioning of the messages in envelopes as $A_x B_xC_x D_x E_x~$, such that a possible envelope-message configuration might be $A_c B_d C_b D_e E_a~$, with message $c$ being slotted in envelope $A$ and message $d$ being slotted in envelope $B$ etc. The correct message for envelope $A$ is message $a$ etc.

I have a potential solution to this problem, but I'm wondering if there are some alternate, simpler solutions.

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astiara
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  • Take a look at the counting formulas here; the recursive formula is especially easy to use when $n$ is small, as it is here. (Make sure that you get the right initial conditions.) – Brian M. Scott Sep 29 '11 at 10:16
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    You're looking for $!5$; see Wikipedia or MathWorld on what's termed 'derangements.' This is arguably a duplicate of this question, and you might also want to see this question or various others like it. – anon Sep 29 '11 at 10:20
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    There is, as the links show, a lot of material on this, of a general character. But for $5$ envelopes, all one needs is patience and carefulness, though a small idea saves quite a bit of time. Letter $a$ goes into one of $B$, $C$, $D$, $E$. By symmetry the number of "good" permutations is $4$ times the number of good permutations that send $a$ to $B$. Now just make a complete list of the good permutations that send $a$ to $B$. Hint: they are of two types (i) $b$ is sent to $A$ and (ii) $b$ is sent somewhere else. It should not take more than a few minutes. – André Nicolas Sep 29 '11 at 14:07

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