Barring leap years and nonuniform birthday distributions, the probability of exactly three having the same birthday is $$\tag1{5\choose 3}\cdot \frac{364^2}{365^4}\approx 0.000075$$ For exactly four, we get $5\cdot \frac{364}{365^4}\approx0.0000001$ and for five $\frac1{365^4}\approx 0.000000000056$, so $(1)$ is also a good approximation for the question about at least three having the same birthday.

Barring leap years and nonuniform birthday distributions, the probability of exactly three having the same birthday is $$\tag1{5\choose 3}\cdot \frac{364^2}{365^4}\approx 0.000075$$ For exactly four, we get $5\cdot \frac{364}{365^4}\approx0.0000001$ and for five $\frac1{365^4}\approx 0.000000000056$, so $(1)$ is also a good approximation for the question about at least three having the same birthday.