Intuition for inclusion exclusion

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.

Re: Intuition for inclusion exclusion

Quote:

Originally Posted by

**terrorsquid** Find the number of solutions for the equation

, where

.

Do you understand how is the number of solutions in which

Re: Intuition for inclusion exclusion

Re: Intuition for inclusion exclusion

Quote:

Originally Posted by

**terrorsquid** Yes, I feel comfortable with the initial step. If only

then

the total number of solutions =

or

... same thing.

Of those have .

So remove that number from

BTW: this is not a problem for inclusion/exclusion.

Re: Intuition for inclusion exclusion

Why do we multiply the second step by two because there are two choices and we don't multiply the first step by two where and are 4?

Quote:

Originally Posted by

**Plato** BTW: this is not a problem for inclusion/exclusion.

My question asks to apply the principles of inclusion-exclusion to solve them :S

Re: Intuition for inclusion exclusion

Quote:

Originally Posted by

**terrorsquid** My question asks to apply the principles of inclusion-exclusion to solve them :S

I have no intention of debating your instructor/textbook.

If the question had asked for or even then clearly inclusion/exclusion would be called for. However, as stated this is a mixed problem.

I have done it in three different ways getting the same answer each time.

Here is a straightforward counting method:

Re: Intuition for inclusion exclusion

Maybe that's why I don't understand what's going on with the inclusion exclusion :D