Stirling's formula

Stirling's formula: [math]\displaystyle{ n! = (1+o(1)) (2\pi n)^{1/2} n^n e^{-n} }[/math].

As a consequence, one has

[math]\displaystyle{ \binom{n}{\alpha n + k} = (c(\alpha)+o(1)) \theta(\alpha)^k 2^{h(\alpha) n} \frac{1}{\sqrt{n}} }[/math]

for fixed [math]\displaystyle{ 0 \lt \alpha \lt 1 }[/math] and bounded k, where

[math]\displaystyle{ c(\alpha) = \frac{1}{\sqrt{2\pi \alpha(1-\alpha)}} }[/math]

[math]\displaystyle{ \theta(\alpha) = \frac{1-\alpha}{\alpha} }[/math]

[math]\displaystyle{ h(\alpha) = \alpha \log_2 \frac{1}{\alpha} + (1-\alpha) \log_2 \frac{1}{1-\alpha}. }[/math]

Thus for instance

[math]\displaystyle{ \binom{n}{n/2 + k} = (\sqrt{\frac{2}{\pi}} + o(1)) \frac{1}{\sqrt{n}} 2^n }[/math]

[math]\displaystyle{ \binom{n}{n/3 + k} = (\sqrt{\frac{9}{4\pi}} + o(1)) \frac{1}{\sqrt{n}} 3^n 2^{-2n/3 + k} }[/math]

etc.

Here is the Wikipedia article for the Stirling's formula.