Yes, but the only way I can think of is much harder to the point of silliness when you have a nice easy way of doing it. This is because it is easy to describe the non-event. There are many many ways to have the event as n gets large.
To use card game naming: You could have just 1 pair with the same birthday, 2 pairs, 3 pairs all the way up to . Then you could have 1 3-of-a-kind, 1 3-of-a-kind plus 1 pair, ....
You would have to add up all the probabilities for the different events, being very careful to never allow the events to overlap.
This is horribly nasty. What's the matter with computing the complement of the event?
There may be some other way to prove it but the standard way is the only general way that I have ever seen (including some quick googleing).