Facebook Friends' Birthdays: A Probability Question

Hey guys,

I have 2,000 Facebook friends and noticed that not one of them has a birthday today (January 10th). I quickly calculated the probability of this as (364/365) ^ 2000 = 0.4% (assuming, for the purposes of this calculation, that each birthday is equally likely).

Here's my question, though: how would I calculate the probability of nobody (out of my 2,000 friends) having a birthday on at least one of the 365 calendar days? In other words, what are the odds that I don't have a birthday on each of the 365 days in the year?

I realize it would be 1-P(birthdays on all 365 days); that said, I have no idea how to calculate P(birthdays on all 365 days) either. :p

Please assume, for the purposes, of this calculation that there are no leap days and that each birthday is equally likely.