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All together then the total paths from \((0,0)\) to \((n,n)\) passing through exactly one of these midpoints is
\begin{equation*}
{n \choose 0}^2 + {n \choose 1}^2 + {n \choose 2}^2 + \cdots + {n \choose n}^2\text{.}
\end{equation*}
in-context