I feel silly asking this as I should be able to work it out, but combinatorics are my enemy.
Consider a collection $x_1, \ldots, x_n$ of real numbers and denote their sum by $s = x_1 + \ldots + x_n$. For $1 \leq p \leq n$ we denote by $ s_p = \sum_{|I| = p} \sum_{i \in I} x_i $ the sum of the elements $x_i$ over all subsets $I \subset \{1, \ldots, n\}$ of cardinality $p$.
Is $s_p = A_{n,p} \, s$ for some integer $A_{n,p}$? Can we write it down?