[Math] Asymptotic for binomial coefficients

Question

Does there exist any asymptotic formula for binomial coefficients ${n \choose k}$ for large $n$ when $k$ is fixed?

Answer 1

For large $k$, when $n$ is fixed, the term ${n\choose k}$ is equal to $0$, since it is equal to $0$ for $k>n$.

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If you fix $k$, then

$${n\choose k} = \frac{n(n-1)\cdots (n-k+1)}{k!}$$

is actually a polynomial (of degree $k$) in $n$, if you expand the expression, since you get $${n\choose k} = \frac{1}{k!}\left(n^k - (1+2+\dots+(k-1))n^{k-1} + \cdots + (-1)\cdot(-2)\cdots (-(k-1))\right)$$