\(\left\{{}\begin{matrix}C_{2020}^k\ge C_{2020}^{k-1}\\C_{2020}^k\ge C_{2020}^{k+1}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\frac{2020!}{k!\left(2020-k\right)!}\ge\frac{2020!}{\left(k-1\right)!\left(2020-k+1\right)!}\\\frac{2020!}{k!\left(2020-k\right)!}\ge\frac{2020!}{\left(k+1\right)!\left(2020-k-1\right)!}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2020-k+1\ge k\\k+1\ge2020-k\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}k\le\frac{2021}{2}\\k\ge\frac{2019}{2}\end{matrix}\right.\) \(\Rightarrow k=1010\)
