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Establish the identity below using a combinatorial proof.
\begin{equation*}
{2 \choose 2}{n \choose 2} + {3 \choose 2}{n-1 \choose 2} + {4\choose 2}{n-2 \choose 2} + \cdots + {n\choose 2}{2\choose 2} = {n+3 \choose 5}\text{.}
\end{equation*}
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