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There is an urn with $M$ numbered balls ($1$ to $M$). All balls are drawn without replacement. Find the probability that at least one ball number matches its pick number.

I tried to find the probability that no ball number matches its pick number and then subtract it from $1$, but couldn't manage to find that either.

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    $\begingroup$ To find the probability no ball number matches, divide the number $D_M$ of derangements of the set $\{1,2,\dots, M\}$ by $M!$. For derangements, please see Wikipedia. There will not be a closed form, but still the answer will look very nice. $\endgroup$ – André Nicolas May 13 '15 at 13:41
  • $\begingroup$ You are welcome. You are likely to bump into ideas connected with derangements again. There are a quite a lot of answers about derangements on MSE, but I think the Wikipedia article is quite thorough, and worth knowing about. $\endgroup$ – André Nicolas May 14 '15 at 1:32

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