How many solutions are there to the equation $w+x+y+z=20$, all are positive integers and $z\leq 4$?

Question

I know how to do the basic type of $w+x+y+z=N$ problem, with non-negative/positive-integer options, but the $z\leq4$ restriction has me stuck. Would you find the possibilities of $z = 1, 2, 3$, and $4$, and then add them up? Is there way some way to use the complement to make this work?

Answer 1

Solve the problem first without the restriction on $z$. Then count the solutions in which $z\ge 5$ by counting solutions to $w+x+y+z'=N-5$, where $z'=z-5$, and subtract these unwanted solutions from the first figure.

This solution and this one show how to deal with this kind of restriction in two similar problems that are just a little more complicated than yours.