0!rganized Emptiness

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.

  1. We can fall back on our old friend, the factorial, and compute 7!
  2. We can use our new friend, the permutation function, and compute \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):

_7P_7 = \frac{7!}{(7-7)!} = \frac{7!}{0!},

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.

Advertisements