Abstract
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.
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