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

*unique objects. This is fantastic, but we don't always want to arrange*

**n***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

*-sized subsets of our*

**r***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 $latex bf{_7P_7}$

Since both expressions count the same thing, they ought to be equal, but then we run into this interesting tidbit when we evaluate (2):

$latex _7P_7 = frac{7!}{(7-7)!} = frac{7!}{0!}&s=2$,

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.