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[EventHorizon]

That is interesting! The ubiquitous e. I suppose, whenever you get a series with a mixture of factorials and exponents, you can always look how to convert the series into f(e^x).

A correction to my post with the birthday (question 2) equation.

Spoiler for B-day equation

The equation to solve is:
N! / ((N-x)!*N^x) = 1/2
or
(N - 1)(N - 2)*...*(N - x + 1) / N^x-1 = 1/2

Yup, that's the equation to start with.

Spoiler for where I went next

Sterling's approximation for factorials.

n! = approx sqrt(2*pi*n)*(n/e)^n

There's a slightly better approximation, but it eventually reduced to Sterling's in my derivation.