Relativism (On the skinning of cats*)
Posted by Mike McClenathan
I chimed in on a thread at College Confidential recently about a probability problem that apparently came from Dr. Chung's book. The first few respondents provided completely legit (but rather technical) explanations of the problem using nCr, and then someone asked whether there was another way. So I jumped in with the way I prefer to solve most counting and probability questions on the SAT: short lists. For the most part, all the solutions offered in the thread were valid.
Why am I posting about that conversation here? Because it underscores an important fact: there are often multiple ways to solve SAT math problems. That's one of the beautiful things about math in general, actually: it's built on itself. That's why you learned addition before multiplication, multiplication before exponents, geometry before trigonometry, etc. My recollection of learning nCr techniques was that they were slowly introduced to us as general solutions to simple combination problems we could solve with simpler counting principles—the kinds of problems you'll see on the SAT.
I get a rush out of breaking a problem down until it's so easy a caveman could do it. That's one reason I'm a pretty good SAT teacher. But you're probably not aspiring to a career in test prep; you're probably just trying to score high enough on the SAT to move on with your life.
So it's completely up to you whether you solve a combination/permutation problem on the SAT simply, the way I like to, or with robust, scalable techniques that would work just as well with many more elements. But I like to remind my students that you'll never need nCr on the SAT; the numbers will stay small. Just like you'll never need trigonometry or logarithms, even though you might once in a while spot a problem that can be solved with them.
At the end of the day, I'm all about information. In my ideal world, you'll know more than one way to solve every problem, so you'll be able to make informed decisions on whether to, say, plug in or do the algebra on a question-by-question basis, instead of being forced into algebra on every question because it's the only way you know.
You've got a math teacher in school 5 days a week who will advocate the mathy way. You've got me, if you want, to show you alternatives. Once you've poked around and seen what's out there, you decide the level of complexity that's most comfortable for you and leads to your best score.