I would like a method to determine the number of positive integer solutions for an linear inequality, of the form:
$Ax + By + ... < Z$ given integer A,B, .. Z and integer $x,y,z,w \ge 0$
For example, there are 11 solutions to $3x + 5y < 15$
I know this is similar to the existing question ( Count the number of positive solutions for a linear diophantine equation ). However, I am unclear about extending it to cover the inequality - Do I need to apply the formula for each 0 .. Z ? Also, it seems difficult to go from even $Ax + By = N$ to $y + z = n$ while remaining in integers.