For a distribution {$p_1,p_2, …,p_m$}, with $p_i>0$ and$\sum_1^m{p_i}=1$ , let $J$ be a subset of {$p_1,p_2, …,p_m$}, size $j$, and $m>j\geq1$. It holds that:
$$\int_0^1\prod_{p_i \in J} (x^{-p_i}-1) dx = \sum_{\text{permutation }(\pi_1,\pi_2,..\pi_j)\text{ of }J}\frac{\pi_1 \pi_2 \ldots \pi_j}{(1-\pi_1)(1-\pi_1-\pi_2) \ldots (1-\pi_1-\pi_2 \ldots -\pi_j)}$$
Is this relation already known and is there a simpler proof than using induction on the size of $J$?