I was given a homework question that is stated in the title. Although I have a conflict with the solution provided, and was wondering if you could help me understand why the solution is correct or if it is indeed incorrect.

Define $X$ to be number of distinct birthdays.

The answer given is to set up a RV $X_i$ which is $1$ if the ith day is a birthday or $0$ otherwise, where:

$P(X_i = 1) = P(\text{at least one person has birthday on day i}) = 1- P(\text{no one has birthday on this day}) = 1 - \frac{364}{365}^{100}$. And so $\mathrm{E}X_i = 1 - \frac{364}{365}^{100}$

Thus $\mathrm{E}X =\mathrm{E}[X_1 + X_2 \dots X_{365}] = 365\left (1 - \frac{364}{365}^{100} \right)$

I think this is incorrect, however. The reason being is that it seems like they are calculating the expected number of birthdays not the expected number of **distinct** birthdays.

The answer that I think is correct is to define $X_i$ as $1$ if the ith day is a ** distinct** birthday and $0$ otherwise. Then:

$P(X_i = 1) = 100 \times \left(\frac{1}{365}\right)\left(\frac{364}{365}\right)^{99}$.

Thus $\mathrm{E}X =\mathrm{E}[X_1 + X_2 \dots X_{365}] = 365 \times 100 \times \left(\frac{1}{365}\right)\left(\frac{364}{365}\right)^{99} = 100 \times \left(\frac{364}{365}\right)^{99}$.

This has been bothering me for quite some time. Any help would be great.