\begin{align} \#\text{Perfect matchings of $K_{2n}$} &= (2n-1)(2n-3)(2n-5)\cdots 5.3.1\\ &=\dfrac{(2n)(2n-1)(2n-2)(2n-3)\cdots 5.4.3.2.1}{(2n)(2n-2)\cdots4.2}\\ &=\dfrac{(2n)!}{(2*n)(2*(n-1))\cdots(2*2)(2*1)}\\ &= \bbox[yellow,5px,border: 2px solid red]{\dfrac{(2n)!}{2^n\times n!}} \end{align}