I have been doing inclusion exclusion questions and while, for the most part, I can see what I am doing in each step and which subsets I am adding/removing, the following type of question I can't seem to see exactly what is happening. I know how to do them, but I am just following a pattern that I have seen from doing a number of them - I can't visualise which subsets I am adding/removing as I do them. I was wondering if someone could tell me what's happening exactly as I do this problem:

Find the number of solutions for the equation , where

So, when and are 4 or greater, then the total number of solutions will be:

I can see what is going on here, every set that includes a value for and less than 4 is excluded. The next steps I do, I get a bit confused what is happening and why...

Suppose: OR < 9 - 4 = 5

Then:

There are two choices so, there are ways this can happen.

Subtract this from the total:

Finally I need to add the instance when both and are < 5 to give the final answer:

I can do these questions, it's just I am doing them without knowing what's going on exactly. So, if the question format was changed up a bit, I might struggle. What has been counted/over-counted at what stage etc.?

Thanks.