Consider a permutation of , and let be the random variable that takes the value if this permutation fixes , and takes the value zero elsewhen. Set .

- Express in terms of for a suitable .
- Find , and hence find .
- Find as , and deduce that converges in distribution to a Poission variable. Of what weight?

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1. Choose fixed points in a set of elements, then choose a derangement of the other elements, we get , so .

2. Let be the number of permutations that fix , by the inclusion-exclusion principle

Then, . Thus, .

3. Clearly . And, by 1., . Thus, converges in distribution to a Poisson variable of rate parameter .

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