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.