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There are $n > 1$ distinct playing cards face-down in front of me. I know what the identities of all the face-down cards are, but I do not know what order they are in. I assign a "guess" as to the identity of each card, making a different guess for each card. Given a number $k <= n$, what is the probability of $X=k$, where the random variable $X$ is the number of cards that I guess correctly?

I thought about this for a while, and every attempt I have made to solve this problem has led to a dead end. Any ideas?

A few observations I have made (which are somewhat obvious): $X$ cannot take on the value $n - 1$, and there is exactly one way to assign the cards such that $X = n$. Despite having this insight, I cannot think of a mathematical statement to succinctly describe this probability.

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This of it like $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right) - P\left(A \cap B\right)$. –  yiyi Apr 26 '13 at 6:46
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This is the rencontres problem (see the answers and the links on the page). –  Did Apr 26 '13 at 7:27

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