I'm not sure I can expand on PEV's hints in a comment, so I'll make it an answer.
You need to know the number of solutions of $$u_1+u_2+\dots+u_r=n$$ when the only restriction on the variables is that they be non-negative integers. Imagine $n+r-1$ dots in a line, and circle $r-1$ of them. The uncircled dots are $n$ in number, and the circled ones divide the uncircled ones into $r$ groups (some of which may be empty), so you get $r$ non-negative integers adding up to $n$. So the question becomes, how many ways can you choose which $r-1$ of the $n+r-1$ dots to circle? Unfortunately, PEV wrote 18-choose-3, where I think what's wanted is 15-choose-3, but now you should see how to get that part of the answer.
Then you ask how to use inclusion-exclusion. It isn't clear whether you mean that you don't see how to get a formula for the size of the union by using inc-excl, or whether you mean that you can write down a formula but don't see how to find the sizes of the $A_i$ and the various intersections that arise, so it's a little hard to help you here. I'll assume it's the second suggestion. So for PEV's $A_1$, let $v_1=y_1-3$, then you have $v_1+y_2+y_3+y_4=9$ and the variables are non-negative, so the previous paragraph applies. Similarly, for the intersection of $A_1$ and $A_2$, let $v_1=y_1-3$ and $v_2=y_2-4$, so you get $v_1+v_2+y_3+y_4=5$ with all variables non-negative.
Can you take it from there?