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Sum of all possible products formed from a set of consecutive integers

  1. Aug 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Say I have a set of consecutive integers up to N, for example up to 5, the set 1,2,3,4,5. I need to know a general formula for the value of the sum of all products that can be formed from a certain number of elements taken from such a set. For example with the 5 set, say I want 3 numbers; I'll get 1.2.3+1.2.4+1.2.5+1.3.4+1.3.5+1.4.5+2.3.4 etc etc. I'd like to find a general formula for any set size and any length of product. Spent most of today looking for one, and from what I did it looks like a nice formula may not exist, but you never know.

    2. Relevant equations



    3. The attempt at a solution

    Let N be the number of elements in the set and S be the number of terms per product. The furthest I got was something like

    [itex]\sum_{i=1}^{N+1-S}i(\sum_{j=i+1}^{N+2-S}j(\sum_{k=j+1}^{N+3-S}k(...\sum_{z=y+1}^{N}z))...)[/itex]

    The final sum (over z here) can be done easily, it gives something depending on y, then the sum over y can be done, and so on, but it very rapidly gets completely out of hand. Any alternative ways of looking at the problem would be greatly appreciated.
     
  2. jcsd
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