$N$ equally able runners enter a race. Each has an assigned shirt number from $1$ to $N$. What is the probability that exactly $m$ runners will have a finishing position equal to their shirt number?
I think the key to the problem would be to find the general probability of none of $N-m$ runners finishing with their numbers, and working from there. However, I'm unable to figure it out. The tricky part is that when you count the number of permutations, it always comes down to a possible situation when the last runner has no choice and you got stuck.