Sum of all possible products of elements taken from couples

Binomial Theorem. In summary, the conversation discusses finding the sum of all possible products of N elements, choosing one from each couple and a certain number of times from the "b" element and the remaining from the "a" element. The suggested solution involves using the Binomial Theorem to calculate the coefficient of x^R in a given expression.
  • #1
Wentu
14
2
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!
 
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  • #2
As an expression I think what you want to do is:

[itex]\Sigma^{N}_{k=0} (a_{k}(\Sigma^{N}_{i=0} b_{i}))[/itex]

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.
 
Last edited:
  • #3
I got the answer from "Michael":
It is the coefficient of x^R in (a0+xb0)(a1+xb1)...(aN+xbN)
 

1. What is the "sum of all possible products of elements taken from couples"?

The "sum of all possible products of elements taken from couples" is a mathematical concept that involves multiplying together all possible combinations of elements from two sets and then adding them together. It is often used in probability and statistics to calculate expected values and in combinatorics to find the number of possible outcomes.

2. How do you calculate the sum of all possible products of elements taken from couples?

To calculate the sum of all possible products of elements taken from couples, you first need to determine the two sets of elements. Then, multiply each element from the first set with every element from the second set. Finally, add all of these products together to get the sum.

3. What is the significance of the "sum of all possible products of elements taken from couples" in mathematics?

The "sum of all possible products of elements taken from couples" is significant in mathematics because it is a useful tool for solving various problems in fields such as probability, statistics, and combinatorics. It allows us to determine the total number of possible outcomes or the expected value of a situation.

4. Can you give an example of how the sum of all possible products of elements taken from couples is used?

One example of how the sum of all possible products of elements taken from couples is used is in calculating the expected value of rolling two dice. The two sets of elements are the numbers on the two dice, and the sum is calculated by multiplying each number on one die with every number on the other die and then adding them together.

5. Is the "sum of all possible products of elements taken from couples" the same as the sum of all possible products of elements taken from two sets?

Yes, the "sum of all possible products of elements taken from couples" is the same as the sum of all possible products of elements taken from two sets. It is simply a different way of referring to the same mathematical concept.

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