Permutations, Combinations, and Why?

We’ve spent a couple of days in precal talking about permutations and combinations.

The first day, we did a fairly straight-forward permutations/combinations lesson. How many ways can you arrange 5 cars in a driveway? How many distinguishable arrangements of the letters in the your full name are there? (Most people have at least one repeated letter.) How many 3-letter “words” can you form if you never repeat a letter? If you only use consonants? If you only use vowels? If I have ten books and want to take 3 on a trip, how many different sets of 3 do I have to choose from?

It was a fairly standard lesson. We’d arrange 2 cars, then 3, then they’d arrange 4 cars, then we’d find the pattern and make predictions. What was slightly different (and what I want to do more of) is this: when we found the pattern, I’d make them figure out why. If they couldn’t, I’d help, but they had to try on their own first.

Why is the number of ways you can arrange 4 cars equal to 4*(the number of ways you can arrange 3 cars)? Why would we divide by the factorial of the number of times a letter is repeated in your name? Repeat with each different topic. When we hit permutations and combinations, I introduced the vocabulary, as well as the notation ${n \choose k}$ and the phrase “binomial coefficient.”

Today, we did Pascal’s Triangle. We started by expanding binomials: $(x+y)^0, (x+y)^1, (x+y)^2, (x+y)^3,$ … . I wrote the coefficients in a triangle, they saw the pattern, and we expanded more binomials. Then I had them compute binomial coefficients, and we arranged them in a triangle. They caught on pretty quickly to the fact that the binomial coefficients we’d computed on Wednesday were, in fact, the coefficients of expanded binomials.

Why would it be true that the coefficient of $x^3y$ in the expansion of $(x+y)^4$ would be the same as the number of ways to choose 3 items out of 4?

I got a lot of blank stares, so I told them to talk about it with their neighbors for five minutes. I wandered the classroom and listened to their discussions (they’re very used to this), and I realized that almost all of them were stuck. After about four minutes, I called it, and we went back to a large group.

One pair of students came up with a pretty good explanation. They had trouble phrasing it, so I helped a bit. Most of them understood it when I explained it, although they said they couldn’t explain it yet themselves. That’s not a total victory, but I had to leave it there for the day.

The most interesting part of class for me, though, was when one girl said, “Ugh! This is math class! I’m not supposed to have to answer why!”

I ask students to explain and justify their answers, although I probably ask that more in Calculus than in Precal. That’s something that I’ll need to work on for next year. But the beautiful thing about math is that there is a why, that things can be proven. Lots of my students want to tell me, “Because that’s just how it is!” or “Because God made it that way!” when I ask them to explain why. That’s true, but that’s not a reason. Math, more than any other subject, always has a reason. I’m not sure how to address this misconception, but it’s something I want to change.