Ross, "A First Course in Probability,7th edition", says the probability that no 3 of n people have the same birthday is "a difficult combinatorial problem", but he gives the following "good approximation".
Suppose we have a trial for each of the triplets i, j, k where and say we have a success if i, j, and k all have the same birthday. Then the number of successes is approximately a Poisson random variable with
If you are looking for an exact solution, Julian Havil in "Nonplussed" refers to a 2004 paper by R.J. McGregor and G.P. Shannon, "On the generalized birthday problem", Mathematical Gazette 88(512):242-48, but Havil doesn't give any details and I don't have a copy of the paper handy.
Havil does say, quoting McGregor and Shannon, that the probability that at least 3 of 88 people will have the same birthday is about 1/2.