Since the thread is locked, I will post here. Using the inclusion-exclusion principle, the probability of at least 1 matching pair is 1 - 1/2! + 1/3! - 1/4! + ... + [(-1)^(n+1)]/n!.
As an aside, as n tends to infinity, this expression tends to 1 - 1/e. So the probability converges to about 0.63212055882.