From easy computation we can get
\begin{align*} {n\choose{\ell}}=\frac{n}{\ell}\cdot\frac{n-1}{\ell-1}\cdots\frac{n-(\ell-1)}{1}\geq\frac{n^{\ell}}{\ell^{\ell}}, \end{align*}
where the last inequality follows from the fact that each of these $\ell$ terms in the product is at least $\frac{n}{\ell}$ and $n\geq \ell>0.$
So, I just wonder if we could do better than this lower bound $\displaystyle\frac{n^{\ell}}{\ell^{\ell}}$ for the binomial coefficient.
Any comments or advice would be appreciated. Thanks for patient reading.
