On the back of the fundamental counting principle, my class has just established the fact that we can use n! to count the number of possible arrangements of n unique objects. This is fantastic, but we don’t always want to arrange all of the n things available to us, which is okay. We’ve also been introduced to the permutation function, which has the very nice property of counting ordered arrangements of r-sized subsets of our n objects. Handy indeed.
Today we made an interesting observation: we now have not one, but two ways to count arrangements of, let’s say, 7 objects.
- We can fall back on our old friend, the factorial, and compute 7!
- We can use our new friend, the permutation function, and compute
Since both expressions count the same thing, they ought to be equal, but then we run into this interesting tidbit when we evaluate (2):
which seems to imply that 0! = 1. To say this is counterintuitive for my kids would be a severe understatement. And in this moment of philosophical crisis, when the book might present itself as a palliative ally, students are instead met with this:
To prevent inconsistency? How in the world are kids supposed to trust a mathematical resource that paints itself into a corner, only tacitly admits such, and then drops a bomb of a deus ex machina in order to save face? I haven’t been so angry since the ending of Lord of the Flies. Especially when this problem appears two pages later:
Okay, 8!. So how many ways can I arrange my bookshelf with a zero-volume reference set? One: I can arrange an empty shelf in exactly one way. And, since we already know that n! counts the ways I can arrange n objects, it follows naturally that this 1 way of arranging 0 things must also be represented by 0!.
There are a lot of good proofs/justifications available for the willing Googler, but this one, to me, seems like the most natural and straightforward for a high school classroom. At a bare minimum, it’s much, much better than, “Because I need it to be true for my own convenience.”
Only a math textbook could take something so lovely and make it seem dirty.