NOTE: I know there are similar questions to this, but the ones on this website are much more complex, and I'd like to get a basic understanding before moving on to them. Please do not mark this as a duplicate.
On a practice quiz:
Use inc-exc to determine the number of solutions to $a+b+c+d = 15$ with $0 \leq a,b,c,d \leq 6$
Here's what I have so far:
Let $A_1$ be the solution to $a+b+c+d=15$ with $a\geq 7, 0\leq b,c,d$
Let $A_2$ be the solution to $a+b+c+d=15$ with $b\geq 7, 0\leq a,c,d$
Let $A_3$ be the solution to $a+b+c+d=15$ with $c\geq 7, 0\leq a,b,d$
Let $A_4$ be the solution to $a+b+c+d=15$ with $d\geq 7, 0\leq a,b,c$
$|A_1 \cup A_2 \cup A_3 \cup A_4| = C(4,1) \cdot C((4+4-1), 4) - \dotsb $
And this is where I get stuck. I found my first term using the logic that I want to count $|A_1|+|A_2|+|A_3|+|A_4|$. How do I continue this to find the rest of the terms? I know what to do after I find the number of non-solutions, just subtract it from the total non-restricted solutions...but I can't figure out how to find the number of non-solutions!
Any help would be greatly appreciated, thanks.