Combinations and Gaussian function I notice that the function $\binom{C}{x}$, where $C$ is some constant, resembles a Gaussian function; for example, here is the plot for $\binom{20}{x}$:

This corresponds to the Gaussian function $a e^{- { \frac{(x-b)^2 }{ 2 c^2} } }$, where $a$ is $\binom{20}{10}$, $b$ is 10, and $c$ (determined through curve fitting) is ~2.2689.
$\binom{20}{x}$ corresponds to $\frac{20!}{x!(20-x)!}$; how is this related to a Gaussian function?
 A: ${20\choose x}*2^{-20}$ is the probability of getting exactly $x$ heads in 20 coin flips.  Or put another way, it is the sum of 20 random variables, each of which has probability $\frac12$ of having the value 0 heads and probability $\frac12$ of having the value 1 head.
The central limit theorem says that the mean of a large number of independent, identical random variables is  (subject to a few conditions) close to a Gaussian distribution.  That is what we have here, the sum of 20 copies of the same distribution.
The central limit theorem is the reason for the ubiquity of the Gaussian distribution in the natural world, in things like people's heights. If you only know three things about probability, the central limit theorem should be one of them.
A: Ignoring normalization, you have noticed an instance of the de Moivre-Laplace theorem (a special case of the central limit theorem) which shows how the binomial distribution can be approximated by a normal distribution for $n$ large.
You have $n=20$ and $p=q =1/2$. 
The theorem tells us the mean for the relevant normal distribution is $n p = 10$ and the standard deviation $\sqrt{n p q} = \sqrt{5}$.
