Ten Letters, Ten Envelopes
A secretary has 10 letters and 10 envelopes, each letter addressed to a different recipient and each envelope labelled for one of them. She places the letters into the envelopes uniformly at random, one letter per envelope (every one of the 10! arrangements is equally likely). What is the expected number of letters that end up in their correct envelope? Give an exact number.
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Source: Montmort's matching problem (problème des rencontres), classic since the early 18th century. Statement written for AxiomIQ.