Find # of Solutions for ΣXi <= C: Combinatorics/Integrals

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SUMMARY

The discussion focuses on determining the number of solutions for the equation ΣXi ≤ C, where Xi are positive integers and C is a constant. The problem is identified as a combinatorial challenge, with references to polytopic numbers as a solution approach. Participants express uncertainty about the intuitive understanding of the connection between the equation's solutions and polytopic numbers, indicating a need for further combinatorial proof to clarify this relationship.

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How do you find the number of solutions for the summation of N variables called Xi, which must be POSITIVE INTEGERS, that are equal or less than some constant number say C.

ΣXi <= C
i = 1,2...N
Xi = 0,1,2,...
I need the number of solutions for this equation

note that "i" is the dummy variable and the subscript for the variable X
do you have to use combinatorics or can you approximate with integrals?
 
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Yeah, this would be a messy combinatorial problem. Lucky for you, someone's already done the hard work. What you want to know about is the polytopic numbers.

The answer is surprisingly simple, though I'm not sure of a good intuitive way to explain why it is. I'm sure that someone else can enlighten us with a neat combinatorial proof.
 
Last edited:
I don't see the connection between the number of solutions to the equation and these
polytopic numbers
 

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