[This is part six in an ongoing series; previous posts can be found here: Differences of powers of consecutive integers, Differences of powers of consecutive integers, part II, Combinatorial proofs, Making our equation count, How to explain the principle of inclusionexclusion?. However, this post is selfcontained; no need to go back and read the previous ones just yet.]
“But”, I hear you protest, “Pi Day was ages ago!” Ah, but I didn’t say Pi Day, I said PIE Day. To clarify:
 Pi Day: a day on which to celebrate the notsofundamental circle constant, (March 14)
 Pie Day: a day on which to eat pie (every day)
 PIE Day: a day on which to explain the Principle of InclusionExclusion (PIE). (That’s today!)
(Actually, I’m only going to begin explaining it today; it’s getting too long for a single blog post!) In any case, the overall goal is to finish up my longlanguishing series on a combinatorial proof of a curious identity (though this post is selfcontained so there’s no need to go back and reread that stuff yet). The biggest missing piece of the pie is… well, PIE! I’ve been having trouble figuring out a good way to explain it in sufficient generality—it’s one of those deceptively simpleseeming things which actually hides a lot of depth. Like a puddle which turns out to be a giant pothole. (Except more fun.)
In a previous post (now long ago) I asked for some advice and got a lot of great comments—if my explanation doesn’t make sense you can try reading some of those!
So, what’s the Principle of InclusionExclusion all about? The basic purpose is to compute the total size of some overlapping sets.
To start out, here is a diagram representing some nonoverlapping sets.
Each set is represented by a colored circle and labelled with the number of elements it contains. In this case, there are 25 people who like bobsledding (the red circle), six people who like doing laundry (the blue circle), and 99 people who like math (the green circle). The circles do not overlap at all, meaning that none of the people who like math also like laundry or bobsledding; none of the people who like doing laundry also like bobsledding or math; and so on. So, how many people are there in total? Well, that’s easy—just add the three numbers! In this case we get 130.
Now, consider this Venn diagram which shows three overlapping sets.
Again, I’ve labelled each region with the number of elements it contains. So there are two people who like bobsledding but not math or laundry; there are three people who like bobsledding and math but not laundry; there is one person who likes all three; and so on. It’s still easy to count the total number of elements: just add up all the numbers again (I get 14).
So what’s the catch?
The catch is that in many situations, we do not know the number of elements in each region! More typically, we know something like:

The total number of elements in each set. Say, we might know that there are 7 people who like bobsledding in total, but have no idea how many of those 7 like math or laundry; and similarly for the other two sets.

The total number of elements in each combination of sets. For example, we might know there are two people who like bobsledding and laundry—but we don’t know whether either of them likes math.
This is illustrated below for another instance of our ongoing example. The top row shows that there are sixteen people who like bobsledding in total, eleven who like laundry in total, and eighteen who like math—but again, these are total counts which tell us nothing about the overlap between the sets. (I’ve put each diagram in a box to emphasize that they are now independent—unlike in the first diagram in this post, having three separate circles does not imply that the circles are necessarily disjoint.) So is probably too many because we would be counting some people multiple times. The next row shows all the intersections of two sets: there are three people who like bobsledding and laundry (who may or may not like math), eight people who like bobsledding and math; and six people who like laundry and math. Finally, there is one person who likes all three.
The question is, how can we deduce the total number of people, starting from this information?
Well, give it a try yourself! Once you have figured that out, think about what would happen if we added a fourth category (say, people who like gelato), or a fifth, or… In a future post I will explain more about the general principle.