An introduction to the “inclusion / exclusion” principle

My older son is enrolled in a math enrichment program that is currently assigning some great counting problems. The program has a 3 week break now so I thought I’d continue on a little bit with some counting ideas and study the “inclusion / exclusion” principle.

We started off last night with the classic case of “derangements” after getting loads of good suggestions on twitter thanks to Patrick Honner, Justin Lanier, Bowen Kerins, and David Butler.

The initial question went like this: If you permute 4 numbers (or snap cubes) how many of those permutations leave no number in its original position?

First we just studied the problem to make sure that we understood it:

Next we moved on to see if we could count all of the cases. Unfortunately our camera ran out of memory in the middle of this video, so sorry for the jump. We did manage to count all of the cases, though:

Having written down all of the cases, we now tried to see if we could count them. The fun thing is that my older son’s idea led us down the “inclusion / exclusion” path:

Finally, I worked through an example with 5 blocks instead of 4. The purpose of this example wasn’t to lay out all of the theory but rather just to reinforce the ideas from the original example of 4 blocks. We’ll build from here in the next project: