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I am having trouble solving exercise 3.4 from E. T. Jaynes' Probability Theory: The Logic of Science.

There are M urns numbered 1 to M, and M balls, also numbered one to M, which are thrown into them, one in each urn. If the numbers of a ball and its urn are the same, we have a match. Show that the probability for at least one match is: $$h = ∑_{k=1}^M{ \frac{(-1)^{(k+1)}}{k!} }$$

How do I solve it?

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The more information you give about what you do understand about this problem, or about how far you have got by yourself, the more likely you are to get a useful answer. –  Chris Godsil Feb 3 '13 at 3:58

1 Answer 1

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The problem of finding the number of ways to do it with no match is the classical problem of counting derangements. For the probability of no match one divides by $M!$.

The Wikipedia article linked to gives a good overview, and includes a proof.

The problem has also been asked and answered repeatedly on MSE.

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Thank you, and I'm very sorry about repeating the question! I wasn't aware of the name "derangement" or the concept. –  Pedro Carvalho Feb 3 '13 at 3:56
    
@PedroCarvalho: No problem! A vast majority of the problems on MSE yield to known results. –  André Nicolas Feb 3 '13 at 6:02

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