Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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


share|cite|improve this answer

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

share|cite|improve this answer

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.