The question I am looking at asks to use a combinatorial argument to show that
$$ \begin{bmatrix} n \\ 2 \\ \end{bmatrix} = \frac{n!}{2} \sum_{m=1}^{n-1} \frac{1}{m(n-m)}$$
I tried to use the recurrence relation for Stirling numbers of the first kind but couldn't find a way to make this work, help appreciated thanks.