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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?

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3 Answers 3

up vote 4 down vote accepted

Yes we can. For each element, we can select $p-1$ out of $n-1$ other elements to form an admissible subset, so the sum is

$$s_p=\binom{n-1}{p-1}s\;.$$

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$A_{n,p} = {n - 1 \choose p - 1} = $ the number of subsets of $\{1, ..., n-1\}$ of size $p - 1$.

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How many times an $x_i$ is counted in $s_p$?

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