In the ‘real world’, there are lots of “how many?” questions where it isn’t practical to be physically counting.  “How many license plates are possible in Missouri [and will that be enough to handle the number of cars]?”  “How many telephone area codes are possible [and with the proliferation of cell phones, will we ever run out?]”
Years ago, I had occasion to enter a Springfield restaurant which unintentionally gave me a miniature version of one of those problems and a rich source of classroom discussions, as well.
The menu had a large and very colorful insert advertising that month’s special.  The insert pictured NINE (9) different options for entrees, and you could “Pick Any Two” for one price.  Elsewhere on the insert was an appetizing picture of 5 different platefuls, and the caption: “Here are 5 of the 81 possible combinations.”
I make a confession:  My immediate gut-level feeling was “There can’t be 81 possible combinations!”  Suddenly, my curiosity was jarred awake, and I HAD to check this for myself. I pulled out the proverbial [paper!] napkin and pen and started scribbling.  I didn’t want to have to count! (And, even so, what if I missed an option?)  I needed a reliable short cut.
Now, it’s true that such a shortcut exists, and I satisfied myself of the answer (one way or the other!), but for the moment, but let’s look closer.
It turns out that this real-world situation makes a wonderful problem with which to grab students and introduce them to this topic of ‘counting methods’.  It also makes a great problem to discuss with teachers as well, for a variety of reasons.
Nine times out of ten, students also want to know what the right answer is (and, of course, whether the restaurant chain made a mistake!). They are willing to discuss various techniques to find the answer (including counting, of course!), make conjectures, share ideas, and other sorts of things that are signs of good problem solving.  And, later, when they are introduced to the various ‘shortcuts’, they marvel at them – and more important, they remember them.  Further, they remember them as helpful shortcuts, rather than formulas-to-be-memorized, presented to them in isolation.
Other good things happen as well.  Often, they begin to ask wonderful questions they don’t normally ask.  Especially those that deal with conditions.  The first time a solver asked “could I repeat an entrée?  May I choose blackened chicken and blackened chicken, for example?”, I was ecstatic.  Partly because of the question itself, yes, but partly because I had to answer “I don’t know – I didn’t ask!”  So, depending on a restaurant’s conditions, there are different solutions possible to the question “How many options are there?”
So – and you saw this coming, didn’t you? – I will temporarily leave the problem with you to have fun with.  What do YOU think?  Was the restaurant chain right?  Are there indeed 81 possibilities, and more importantly, can you give an argument for your own answer? Are there other conditions?  I’d be delighted to hear any of your thoughts.  And we’ll summarize and tie up loose ends in a future column.
One final reminder.  What’s going on above is what mathematics is about.  More than moving decimal points or factoring polynomials, math is about solving real-world problems.  Happy Counting!