Birthday Paradox: why permutations and not combinations?

Question

The Birthday Problem: given $n$ people (typically $n<365$), what is the probability that some pair of them share a birthday (omitting Feb 29th, for simplicity)?

The solution: First, find the probability that all $n$ people have different birthdays. Here is where I am confused. The solutions I have seen all say this probability is: $$ \frac{_{365}P_n}{365^n} $$

Why isn't this $_{365}C_n$,instead?

Answer 1

If you did combinations, you would basically choose the birthdays but not assign them to the $n$ people. However the denominator presumes the birthdays have been assigned (365 choices for Alice, 365 choices for Bob, etc.)

Answer 2

The probability with $n$ people is, by considering the first, then the second, then the third, ..., $$\frac{365}{365}\times \frac{364}{365}\times \frac{363}{365}\times \cdots \times\frac{365-n+1}{365}.$$

Now simplify.