Probability of 3 people in a room of 30 having the same birthday
The birthday paradox is a common problem about the probability that any 2 people from a random set of $N$ people share any common birthday. My question is a generalisation: What is the probability that $M$ people from a random set of $N$ people share any common birthday. The only difference is generalising 2 to $M$.
I'm particularly interested in the case where $M=5$ if that makes it significantly easier to solve for.