The Taylor series expansion of the exponential function (the constant e ≈ 2.718281828) 1. e x = 1 + x + x 2 2 ! + ⋯ {\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+… See more

The argument below is adapted from an argument of Paul Halmos.[nb 1] As stated above, the probability that no two birthdays coincide is 1. 1 − p ( n ) = p ¯ ( n ) = ∏ k = 1 n − 1 ( 1 − … See more

Arbitrary number of days
Given a year with d days, the generalized birthday problem asks for the minimal number n(d) such that, in a set of n randomly c… See more

First match
A related question is, as people enter a room one at a time, which one is most likely to be the first to have the same birthday as someone al… See more