If there are 100 people in a room then what is the probability that at least 2 of them share birthdays? I read that answer should be $1-\left(\frac{365!}{265!\times365^{100}} \right)$, and I understood why it is so.
However if I solve it another way I am getting the wrong answer. If we think of the problem as assigning birthdays to people, then there are ${365 \choose 100}$ ways to assign each person with a different birthday and ${464 \choose 100}$ ways so that one birthday could be assigned to more than one person. So why the following equation gives me the wrong answer: $$1-\frac{{365 \choose 100}}{{464 \choose 100}}$$
I don't understand why the order of people matters in this question.