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

[qbslug]

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?

 

[Manchot]

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.

 

[qbslug]

I don't see the connection between the number of solutions to the equation and these
polytopic numbers