3.1.1 problem 1

If the $k$-th person’s arrival results in the first birthday match, the first $k-1$ people have $365\ast 364\ast \cdots \ast (365-k+2)$ choices of birthday assignments such that no two people have the same birthday. The $k$-th person has $k-1$ choices of birthdays, since their birthday must match that of one of the first $k-1$ people.

Thus,