Cancellation Effect amongn-Body Clusters in LiquidHe3and Nuclear Matter

We prove a simple combinatorial theorem regarding the cancellation between direct and exchange many-body clusters in liquid ${\mathrm{He}}^3$. Under certain simplifying assumptions, our result would mean that all $n$-body clusters cancel each other for $n>~3$. Deviations of the real ${\mathrm{He}}^3$ system from these simplifying assumptions are also discussed. Corresponding statements are made throughout for the nuclear-matter system as well.