The Perfect Combinatorics Problem

In this post, I’m going to extol the virtues of my favorite combinatorics problem.  You’ve probably heard it, or some version of it, before:

A pizza parlor offers ten different toppings on their pizza.  How many different types of pizza are possible to make, given that a pizza can have any number of toppings, or no toppings at all?

Just in case you aren’t familiar with this problem and want to work it out for yourself first, I’m putting most of this post after the jump.  First, a shout out: I remember doing this problem with Michigan State Professor Bruce Mitchell, who used to teach Saturday-morning math enrichment classes at my middle school, and whose enthusiasm and humor kept me coming back.  Second, some pizza:

http://www.neatorama.com

You may prefer to pretend you never saw that.