It was my mate's birthday yesterday and he noticed that several other people we know also had their birthdays on the same day. We began discussing the Birthday Problem and attempted solutions for more than 2 people and tried to generalise it, but we couldn't find a satisfactory solution for an arbitrary number of people. So:(adsbygoogle = window.adsbygoogle || []).push({});

For a personand a set of peopleAwith cardinalityS, assuming there are 365 days in a year (i.e ignore leap years) and that birthdays follow a discrete uniform distribution, what is the probability that exactlynpeople inmhave the same birthday asS?A

I've read through the ideas discussed here: https://www.physicsforums.com/showthread.php?t=664296 but can't see how to extend it to an arbitrary number of people.

I've also seen similar problems such as section 2.4 in: http://www.math.ucdavis.edu/~tracy/courses/math135A/UsefullCourseMaterial/birthday.pdf

where the probability that at least 1 set of 3 people share a birthday is calculated. Notice that this solution is undefined if|>365 so there are several subtleties to the problem.S|

Any ideas?

Note: It's been the best part of a decade since I've studied probability.

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# Generalisation To The Birthday Problem

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