Triple Birthday Collision Threshold

In a room of $n$ people with birthdays uniform on $\{1,\ldots,365\}$, let $A$ be the event that at least three people share the same birthday.

(a) Using a Poisson approximation (model the occupancy of each day as an independent $\text{Poisson}(n/365)$ variable), derive an approximate formula for $P(A)$.

(b) Find the smallest $n$ such that $P(A) \ge \frac{1}{2}$ under this approximation.