the authors are basically rediscovering Chen-Stein’s concept of Poisson approximation... which can be used to almost trivially approximate any birthday problem variant. For example, how many people do you need to have a 50% Probability that at least 6 people were born on the same day on Jupiter? Ez w/ Poisson approximation. I first learned about this in Persi Diaconis’ graduate probability class but the method is very simple

https://projecteuclid.org/euclid.ss/1177012015