If it is stool #1, then stool #2 must be empty; this then reduces the problem to a new problem of looking at a row of 8 stools of which two are to be occupied by non adjacent customers.

If it is stool #2, then stool #3 must be empty; this then reduces the problem to a new problem of looking at a row of 7 stools of which two are to be occupied by non adjacent customers.

If it is stool #6, then stool #7 must be empty; this then reduces the problem to a new problem of looking at a row of 3 stools of which two are to be occupied by non adjacent customers.

You should see that the two customer problem is easier than the original. You probably have learned about combinations; 8C2 is the number of ways to choose 2 objects from 8 identical objects. Why is 8C2 - 7 relevant to the first step above?

The key to these types of problems is to imagine a picture, or several pictures.

For example, I imagine that the stools are essentially in a row or at least not two seats are 'the same' in terms of the complete layout of the cafe. (For comparison, if they were saying 10 seats around a round table the process would be different, in detail).

The next step would be to imagine certain examples. can you see some that are 'the same' and some that are different.

e.g. calling the people a,b,c, and empty seats x:

xxxaxbxcxx and xxxbxaxcxx are the same,
however,
xxxaxbxcxx and xxaxxbxcxx are different.

Now this particular problem, with rules on not sitting beside one another, gets complicated. I know the method, but it may not immediately be natural to you.

First, I will stop considering the three people as different, but will use the same letter for all three.

xxxlxlxlxx would be the first example, using l for each of the people.

We are looking at ways to place three l's around the seven x's, so that no two l's are side by side. I am now going to change the picture a little.

Put down the seven x's with possible gaps for people.

x x x x x x x
There is only one way to do that! Now think of putting in the three people. Where can they go? They can either go between two x's OR on either end. There are 8 possible places for the first person to sit. The next person cannot sit beside the first, so they have fewer place - 7. etc.

Finally, it did not matter who sat first or second. You really just wanted to grab 3 of the eight possible spots. This is, in fact, a 'combination' problem (as you may already have done) where you pick three, order does not matter.

Notice that I used pictures to organize my playing around. I played with examples and simple pictures, to see what mattered and what didn't. I strongly encourage doing that with all these problems. When we get good at these problems, sometimes we just do the pictures in our heads, and people don't notice that we are using them.

Finally, I tried filling things in order, then checking whether the order mattered. In the end, I could NOW put down a simple formula - but that is not something you (or I) should expect to do having just read the problem.

There are some basic principles around which these counting problems are organized. Things like a sequence of actions, leading to a multiplication. Things like 'equivalence' (how many actual seating patterns are considered the same - or equivalent? This leads to a division).

In fancy problems, people use something called Polya's counting principle and group theory. You will probably not study this in your course, but it is comforting to know that there are deeper principles and everything is not just done with your bare hands anew each time!