I want to upper-bound the following Binomial coefficient with a Polynomial $p(n)\in O(n^c)$: $$ {n \choose k}=\frac{n!}{k!\cdot\left(n-k\right)!}\leq p(n) $$ I know that $k\leq n$ but which $p(n)$ I can always use here?
EDIT: The question is coming from the following case:
$$ {n\choose n-17}\leq n^{17} $$ but for some reason you can't upper-bound ${n \choose 0.5n}$. why?