Kitten complained that some math programs keep repeating the same kind of problems over and over, with bigger numbers: “They don’t get any harder, they just get longer. It’s boring!”
So we pulled out the Counting lessons in Competition Math for Middle School. [Highly recommended book!] Kitten doesn’t like to compete, but she enjoys learning new ideas, and Batterson’s book gives her plenty of those, well organized and clearly explained.
Today’s topic was the Fundamental Counting Principle. It was review, easy-peasy. The problems were too simple, until…
Pizzas at Mario’s come in three sizes, and you have your choice of 10 toppings to add to the pizza. You may order a pizza with any number of toppings (up to 10), including zero. How many choices of pizza are there at Mario’s?
[The book said 9 toppings, but I was skimming/paraphrasing aloud and misread.]
- Can you figure out the answer?
Heading Down the Wrong Path
Conditioned by the easy problems earlier in the lesson, Kitten popped out an answer very quickly. She multiplied the number of crust options times the number of toppings, for different pizzas. (This teacher makes the same mistake!)
If I had been thinking well myself, I would have encouraged her that this is the correct answer if we wanted the number of one-topping pizzas.
As soon as I said that answer was wrong, she realized there had to be more options. She started thinking about permutations: 3 choices for the crust size (she knew that was correct), times 10 choices for the first topping, times 9 choices for the second topping, times…
I Always Say Too Much
Wanting to save her from the frustration of a nearly-endless calculation that I knew would end in another wrong answer (and possibly induce an emotional melt-down), I spoke up: “Sometimes, when a problem is too tough, a good thing to try is to make it simpler. What could—“
Immediately, Kitten dropped her marker and hunched down over the whiteboard with eyes clenched tight, both hands over her ears.
I took the hint and shut up.
We sat there.
Finding the Right Road
After what seemed like forever (but was probably less than five minutes), Kitten picked up her marker and started to make a list:
“With two toppings, we get 5 pizzas.”
I pointed out that most pizza places have their own rules about which order the toppings go on, so they wouldn’t serve both Cheese-Pepperoni and Pepperoni-Cheese.
She agreed: “The cheese should always go on first, under the other toppings. That way you can pick off whatever you don’t like.”
What Other Toppings Shall We Try?
Her next topping was ham (H), and she had no trouble finding the 8 combinations for a 3-topping option.
We were momentarily stumped on a fourth topping — peppers or pineapple wouldn’t work, since we already had a P, and mushrooms were unthinkable. Kitten decided to use O. (Onion? Olives? Olives will definitely get picked off.) It was trickier to make sure we had all the possibilities, but in the end she listed 16.
“Each new topping makes it twice as many,” she said. “So there are .”
That made sense, because every time you add a new topping, you can still order all the options you had before, or you could make each of those choices with the new topping added to it. Double your pleasure!
And having figured out earlier that (and taken enough delight in the number’s self-referential pattern to remember it), Kitten easily calculated the final answer: Mario can make 3,072 different pizzas.