TITLE: A question about limit $\lim_{n\to\infty} {n \choose n/2}/2^n$
QUESTION [7 upvotes]: My question is: What is the result of this limit: 
$\displaystyle \lim_{n \to +\infty} \frac{{n \choose n/2}}{2^n}=$ ?

REPLY [9 votes]: For even $n$, this is the probability that you flip $n$ fair coins and get exact $n/2$ heads.  
As shown by the Central Limit Theorem, this binomial distribution can be asymptoticly approximated by a Gaussian distribution of mean $n/2$ and variance $n/4$ for large $n$, and this point will be close to the highest density so $\dfrac{{n \choose n/2}}{2^n} \approx \sqrt{\dfrac{2}{\pi n}}$, which falls towards $0$ as $n$ increases.