Function defining $nCr$

Question

While playing with my calculator I found that surprisingly, it is giving values for fractional values of $n$ and $r$

How is that possible?

Answer 1

This is tied to the notion of factorial and how it is generalized to non-integer values. As I'm sure you're aware, the factorial is defined for a nonnegative integer $n$ by

$$n! = n(n-1)(n-2)\cdots(2)(1)$$

There is a generalization to this, the gamma function $\Gamma(n)$, to non-integer values. That is, $\Gamma(n+1) = n!$ whenever $n$ is a positive integer, but its definition will let you calculate its value for non-integer values. Namely, that definition is

$$\Gamma(n+1) = \int_0^\infty e^{-t} t^n dt$$

You can read up more on that function here. This generalization works for all complex numbers as well, except for the negative integers. (The negative integers are poles for $\Gamma(n)$, and thus the function is undefined there.)

Your calculator probably uses methods to approximate that integral, and return a value for the function.