\[\begin{array}{l}
{\text{Докажите тождество:}} \hfill \\
\sum\limits_{k = 0}^n {\frac{{\left( {2n} \right)!}}{{{{\left( {k!} \right)}^2}{{\left( {\left( {n - k} \right)!} \right)}^2}}}} = {\left( {C_{2n}^n} \right)^2}. \hfill \\
\end{array}\]
\[{\left( {C_{2n}^n} \right)^2} = \frac{{\left( {2n} \right)!}}{{{{\left( {n!} \right)}^2}}} \cdot \sum\limits_{k = 0}^n {{{\left( {\frac{{n!}}{{\left( {n - k} \right)!k!}}} \right)}^2}} = \sum\limits_{k = 0}^n {\frac{{\left( {2n} \right)!}}{{{{\left( {k!} \right)}^2}{{\left( {\left( {n - k} \right)!} \right)}^2}}}} \]
