I think you could proceed in this way: $P(at \ least \ 3) = P(at \ least \ 2) - P(exactly\ 2)$
since $P(at \ least \ 2) = 1 - P(no \ one)$
thus $P(at \ least \ 3) = 1 - P(no \ one) - P(exactly\ 2)$
where
$$P(no \ one)=\frac{365!}{(365-n)!365^n}$$
and
$$P(exactly\ 2)=\binom{n}{2} \frac{365!}{(365-n+1)!365^n}$$
for the derivation of the last equation the ingredients are as follow:
- Select a pair, the probability that they share the same birthday date is: $\frac{1}{365}$
- The probability that the remaning "n-2" people have different birthday date is:
$\frac{364\cdot363\cdot...\cdot(364-n+3)}{365^{n-2}}$
- As to select a pair there are $\binom{n}{2}$ choices
for the pair, the totalprobability is given by the following product:
$$\binom{n}{2} \frac{1}{365} \frac{364\cdot363\cdot...\cdot(364-n+3)}{365^{n-2}}=\binom{n}{2} \frac{364!}{(364-n+2)!365^{n-1}}=\binom{n}{2} \frac{365\cdot364!}{(364+1-n+2-1)!365\cdot365^{n-1}}=\binom{n}{2} \frac{365!}{(365-n+1)!365^n} \ \square$$
references
Probability question (Birthday problem)