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?

• It's a bit hard to imagine that this question hasn't come up before, but I did not look very hard for it.
– MJD
Commented May 24, 2012 at 4:30
• Agree, Mark. It is also covered in all the textbooks. Yet, it is possible to see this on you own without really studying the topic. I recall a certain schoolboy collecting data on the sum of the rolls of 5 dice... Commented May 24, 2012 at 4:37
• Sure! I just meant that there might already be an answer on this site that is better than my answer, and we should link to it.
– MJD
Commented May 24, 2012 at 4:38

${20\choose x}*2^{-20}$ 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.
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 $\sqrt{n p q} = \sqrt{5}$.