How many combinations have never been chosen at the ice cream shop? 

 To follow up a previous  post  of mine, here's another statistics-related lesson to do with your kids.  I came up with it at an ice cream shop with my 10-year-old daughter a couple of weeks ago.  The point of the lesson is about the power of combinatorics and really, really big numbers.  The result is pretty surprising.  Here's the recipe: 
 


 INGREDIENTS: An ice cream shop, some money for some ice cream, a kid, and a calculator.  [I hear 2 objections. To the first: Don't worry, you're probably already carrying a calculator; look closer at your cell phone.  The second is: shouldn't we be requiring kids to make the calculations themselves?  The fact is that lots of famous mathematicians and statisticians are pretty bad at arithmetic, even though they are obviously spectacularly good at higher level mathematics.  Being able to multiply 2-digit numbers in your head is probably useful for something, but understanding the point of the calculation -- why you're doing it, what the inputs are, and what the result of the calculation means -- is far more important.] 

 DIRECTIONS: Make your order, sit down, and, while you're eating, pose this question to your kid: Suppose the choices on the menu on the wall have never changed since the shop opened.  How many choices do you see that have never been chosen even once? 

 After thinking about weird but fun options like pouring coffee in an ice cream cone, we try it a little more systematically. So we first set out to figure out how many options there are.  So I ask, "how many ice cream flavors are there?"  My daughter counts them up; it was 20.  So how many combinations of one flavor can you have?  20 obviously.  How many combinations of two flavors can you have (where for simplicity, we'll count a cone with chocolate on the bottom and vanilla on the top as different from the reverse)?  The answer is 20 x 20 or 400.  (Its not 40, its 400.  Think of a checkerboard with one flavor down the 20 rows and another across the 20 columns and the individual squares as the combination of the two.) 

 So how many toppings could we have on that ice cream?  She went to the counter and counted: 18. And then did 18*400, which she figured out is 7,200. After that we used the calculator and just continued to multiply and multiply as I point out categories on the menu and she counts each up.  The total gets big very fast.  We got to numbers in the  trillions  in just a few minutes. 

 So we find that the total number of options is a really big number.  But what does that say about how many options have been tried? 

 Let's suppose, I say, that it only takes one second for someone to make their choice and receive their order, and that the shop is open 24 hours a day, 7 days a week, all year round.  (You could make more realistic assumptions, and teach some good data collection techniques, by watching people get their orders and timing them.)  Then we figure out how long it would take for the shop to have been open (under these wildly optimistic assumptions) in order to serve up all the options.  To calculate the number of years, all you do is take the number of options, divide by 60 (seconds a minute), 60 (minutes an hour), 24 (hours in a day), and 365 (days a year).  In our case, to serve all the options, the shop would have had to be open for around 43,000 years! 

 So even if the shop had been open for 100 years, it couldn't have served even a tiny fraction of the available options.  So how many choices have never been tried at the ice cream shop?  Its not just the few that we can cleverly dream up.  In fact, almost all of them (over 99 percent of the possibilities) have never been tried! 
(At which point my daughter said, "ok, let's get started!") 

 Actually, if you go to a deli and try this, you can get much larger numbers.  For example, if the menu has about 85 items, and each one can be ordered in 10 different ways, the number of possible orders (10 to the 85) is larger than the number of elementary particles in the universe.