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Counting Strategies

Counting strategies. In this lesson, we will simply review all the counting strategies that we’ve studied in this module. First of all, remember that in counting problems, and means multiply, and or means add. That’s a very simple but big idea.

Remember that listing possible outcomes will rarely solve the problem, but doing so can give you some idea about how to proceed. So it can give us a sense of which method we’re going to use. If you can break the problem into stages, then you can apply the fundamental counting principle. Remember to count the most restrictive stage first.

We can put n different items in order in n factorial different arrangements. If some of the elements are identical, we divide by the number of identical elements. In general, if our counting approaches produces repetitions, we divide to eliminate repetitions. Remember to use combinations when order doesn’t matter.

Remember that we have we have multiple ways of actually calculating the combinations. Remember that permutations are just another way of talking about the fundamental counting principle. And it’s much easier to solve those using the fundamental principle than learning a new formula.

In general, throughout all math on the test, knowing formulas is not a big part of what you need to know. In fact, knowing formulas constitutes probably about 10% of what you actually need to know to be successful on the test. In counting problems, in combinatorics, knowing the formulas alone is about 2% of what you need to know.

In other words, if you know the counting formulas and that’s all you know, you know almost nothing. You need to know things like the fundamental counting principle, counting with, using restrictions, what if there are repetitions, all the things that we’ve talked about in this module. These are more important than memorizing formulas.

A very left-brain approach to the quant section is to memorize formulas and rules and procedure. That approach will yield only limited success in counting problems. As discussed a few times in these videos, counting is a much more right-brain area of mathematics. The left-brain question is, what is the right thing to do in this problem?

The right-brain question is, what is the right way to look at this problem? What is the right way to frame the information? What are the right perceptual choices to make in order to dissect the problem? More so than in any other math problems, it’s important to read solutions to all counting problems. Even if you get them right, read the solution or watch the explanation video.

When you read these solutions, don’t simply focus on what to do because then you’ll be left with that question, why did they do this instead of that? Instead, focus on this, how the solution framed the problem, noticing the perceptual choices, the, the solution made at the very outset. As you study solutions in this way, you observe patterns, and it’s by learning these patterns that you develop a deep understanding of combinatorics.

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