I'm trying to make an efficient algorithm to find how many solutions there are to the equation $$Ax+By+Cz=D$$ where $A,B,C,D\in \mathbb Z$ and the range for $x,y,z\in \mathbb Z$ are given by the user.
Is it correct to use stars and bars for this? I don't think I've seen a stars and bars question where the variables had coefficients.
It's not only about variables having coefficients. To use "stars and bars" the range of the variable should be a subset of the set of non-negative integers(since each variable represents a count of something ) and all the coefficients should also be non-negative. In that case, you can use "stars and bars" or "Inclusion-Exclusion" or any other combinatorial machinery to solve these problems.