Find the probabilities for the various configurations of the birthdays of $30$ people born in a leap year.
My guess is that total configurations are $\binom{366}{30}$. Is that right? These are for choosing the days, assigning them to the $30$ people can be done in different ways or will that be taken care by the combination itself?
If the persons are distinguishable by their identity then - as commented - there are $366^{30}$ configurations.
If the persons are only distinguishable by their birthdays then you must find the number of tuples: $$\langle a_1,\cdots,a_{366}\rangle\in\mathbb Z_{\geq0}^{366}$$ that satisfy:$$\sum_{i=1}^{366}a_i=30$$
Here $a_i$ corresponds with the number of birthdays on day $i$.
This can be done with application of stars and bars and gives outcome $$\binom{30+366-1}{366-1}$$