Only a gathering of 23 people are needed to to make the probability of two of them having the same birthday greater than 50%. This is a well known and surprising result.
In deriving the expression for probability of birthday coincidence in a group of n people, the approach seems always to consider the probability of non-coincidence:
p 364/365 x 363/365 x 362/365 ......x (365-n)/365
n 1 2 3 n
and subtract this from 1 to find p' (coincidence).
Is there a way of deriving the formula by considering the probability of coincidence directly and not having to go via the non-coincidence route?