\prod\limits_{i = 0}^{n/2} {\left( \begin{array}{c}
 n - 2i \\ 
 2 \\ 
 \end{array} \right)} = \frac{{n!}}{{2!\left( {n - 2} \right)!}} \cdot \frac{{\left( {n - 2} \right)!}}{{2!\left( {n - 4} \right)!}} \cdot \ldots \cdot \frac{{4!}}{{2!2!}} \cdot \frac{{2!}}{{2!0!}} = \frac{{n!}}{{2^{\frac{n}{2}} }}