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

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To find the number of solutions for the summation of N positive integer variables Xi that are less than or equal to a constant C, combinatorial methods are typically employed. The problem is recognized as a complex combinatorial challenge, but it has been addressed through the concept of polytopic numbers. While the exact intuitive explanation may be elusive, the relationship between the solutions and polytopic numbers is noted as significant. The discussion suggests that there may be a combinatorial proof that could clarify this connection. Overall, the exploration of this problem highlights the intersection of combinatorics and integrals in solving such equations.
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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.
 
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I don't see the connection between the number of solutions to the equation and these
polytopic numbers
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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