Suppose we have created an army of n clones which are completely identical(except they may have different birthdays). The cloning happened at different times such that all 365(disregarding the 366th day) birthdays are equally likely.
What is the probability of at least 2 people sharing a birthday in this indistinguishable setting.
In the original birthday problem the solution is $$1-\frac{365Pn}{365^n}$$ But here the solution assumes the distinguishability of the people.
Also let us note that for the indistinguishable case the solution $$1-\frac{365 \choose n}{365+n-1 \choose n}$$ is incorrect because it fails to regard the probability weighting of each outcome as they are not equally likely(for example probability of two people having two Sep 1s is less that probability of having Sep 1 and Sep 2 as it can happen in two cases).