# Seemingly Simple Probability Problem

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?

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.
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