Here's a nice probability puzzle I have thought about for a class I'm TAing, I'm curious to see different solutions :) It goes like this:
We have a classroom with $n$ seats available and $m \leq n$ incoming students. Each student has an (ordered) list of $k \leq n$ preferences for the seat he is going to take, where $k$ is some fixed positive integer. If at the moment of his arrival, a person's $k$ favorite seats are already taken, then he randomly chooses a seat from the remaining $n-k$. What is the probability that everyone occupies one of his favorite $k$ seats?