Number of non-negative integer solutions for linear equations with constants How do we find the number of non-negative integer solutions for linear equation of the form: 

$$a \cdot x + b \cdot y = c$$

Where $a, b, c$ are constants and $x,y$ are the variables ?
Not a complete answer, but a relatively simple one and approximate one. By Schur's theorem of combinatorics?, the number of solutions is asymptotically ($c \to \infty$):

$$ \frac{c}{ab} $$



Schur's theorem of combinatorics states that the number of solutions of (with $a_i$ relatively prime):

$$ \sum_{i=1}^M a_i x_i = c $$

is:

$$ \frac{c^{M-1}}{(M-1)!\prod a_i} $$



? This name is used by Wilf's Generatingfunctionology, but I cannot seem to find it elsewhere. It appears that Schur has many theorems.
You might be looking for Diophantine equations.

Check this link or this for an explanation!