Escamillo, I am sorry but your reasoning here is faulty. The question is, what is the probability that in a group of 23, AT LEAST two share the same birthday. Note that the question is not (or should not be) what is the probability that exactly two people share the same birthday?

The only way to calculate this is to calculate the probability that everybody has a different birthday. Then you subtract that probability from 1 and you have the probability that evrybody in the group does NOT have a different birthday. That is another way of saying that at least two members of the group share ethe same birthday

QED

Murray