Combinatorics

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Stirling's Approximation and 2n Choose n

COMBIN-QQEOYG

Stirling's approximation says that $n!$ is asymptotically equal to $\sqrt{2\pi n}\left( n/e\right)^n$.

This means that the ratio:

$$\frac{n!}{\sqrt{2\pi n} \left(\frac{n}{e}\right)^n}$$

...approaches 1 as $n$ goes to $\infty$. We write:

$$n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$$

Which expression is asymptotically equal to the binomial coefficient ${2n\choose n}$?

A

${2n\choose n} \sim 2\sqrt{2\pi n} \left(\cfrac{n}{e}\right)^n$

B

${2n\choose n} \sim\sqrt{2\pi n} \left(\cfrac{n}{e}\right)^{2n}$

C

${2n\choose n} \sim2\sqrt{2\pi n}\,2^n$

D

${2n\choose n} \sim\cfrac{2^n}{\sqrt{2\pi n}}$

E

${2n\choose n} \sim\cfrac{4^n}{\sqrt{\pi n}}$