# Sum of all possible products of elements taken from couples

by Wentu
Tags: combinatorics, couples, elements, products
 P: 10 Hello I have N couples of real numbers higher than 1. Let's call them like (a0,b0), (a1,b1),...,(aN,bN) I have a number R <= N. I need the sum of all the possible products of N elements, chosing one from each couple but exactly R times the "b" element and N-R times the "a" element. Which is the best way to do it? As an example: (2,3), (5,7), (11,13) N = 3, R = 2 I need 2x7x13 + 3x5x13 + 3x7x11 Thank you!
 P: 1,226 As an expression I think what you want to do is: $\Sigma^{N}_{k=0} (a_{k}(\Sigma^{N}_{i=0} b_{i}))$ I have no idea if there is any way to compute this other than just doing it. EDIT: Nevermind, I see you don't want "sum of all possible products of N+1 elements" but sum of all possible products of a choice of R elements from the N+1 elements. No idea, you're probably going to have to write a program for that.
 P: 10 I got the answer from "Michael": It is the coefficient of x^R in (a0+xb0)(a1+xb1)...(aN+xbN)

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