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Let $F$ be the number of fixed points of a random permutation on $n$ items. Show that as $n$ approaches infinity, the distribution of $F$ approaches a Poisson distribution with a mean $(\lambda)=1$.

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In fact $F$ and the Poisson distribution have the same moments up to and including the $n$'th. See e.g. groups.google.com/group/sci.math/browse_thread/thread/…? –  Robert Israel May 4 '12 at 6:30
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