Shared birthdays in a room of N people

Assume 365 equally likely birthdays and ignore leap years.

(a) For a room of $N\le365$ people, find the probability that two or more of them share a birthday.

(b) You are one of $N$ people in a room. Find the probability that at least one of the others has your birthday.

(a) $1-\dfrac{365!}{(365-N)!\,365^N}$; (b) $1-\left(\tfrac{364}{365}\right)^{N-1}$. In both cases compute the complement. (a) All birthdays distinct has $365\cdot364\cdots(366-N)$ favourable assignments out of $365^N$. (b) Each of the other $N-1$ people independently avoids your birthday with probability $\tfrac{364}{365}$.