 Thus,

Then, replace with non-negative integers for each term in the formula of until , for example, when ,

The first expression derived for can be approximated as

Therefore,

An even coarser approximation is given by

which, as the graph illustrates, is still fairly accurate.According to the approximation, the same approach can be applied to any number of "people" and "days".If rather than 365 days there are , if there are persons, and if , then using the same approach as above we achieve the result that if is the probability that at least two out of people share the same birthday from a set of available days, then