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Introduction to Counting
The introduction to the counting module. So as you’ve been looking down the list of modules, this one might have been puzzling to you because, of course, everyone knows how to count. Of course, the test is not gonna ask you, you know, something like count from one to 20. The math section, of course, is not gonna ask you that.
This kind of counting measured on the test concerns problems such as in how many ways can four novels and three reference books be arranged on a shelf so that blah, blah, blah, that the, some kind of condition about the, the novels are on the outside, the reference books on the inside, or different orders, this sort of thing. So it’s about looking at orders and sets and combinations of things. That’s what we’re talking about when we talk about counting.
So we will discuss counting the number of ways to select or arrange a large number of items. Technically, this branch of math is called combinatorics. That’s kind of a fancy scary word ,so we just use the simpler word counting. Technically, what we’re learning in this module though is elementary combinatorics. In this lesson, we will just discuss a few introductory points.
The first concerns the seemingly simple words and and or. Now, those look like words from ordinary conversation, but in fact, they are mathematical objects with mathematical meaning. In counting, and as we will see in probability, and means multiply, and or means add. That’s a really big idea right there.
So, let’s think about this a little bit. If, at a fancy dinner, you could choose one of three meat entrees or one of two vegetarian entrees, how many choices of entree would you have? Well, these are the entrees. So you have the, any of the three meats or any of the two veggies.
So all together, what we have are five choices, 2 plus 3 equals 5. So the or meant that we add the 2 and the 3. By contrast, if you could choose one of three options, now we don’t care whether they’re veggie or meat or whatever, but we have three entrees and we can pick one of two desserts to go with it, how many entree plus dessert combinations are there now?
So now we have three entrees. With any one of them, we could pick dessert number one. Or with any one of them, we could pick dessert number two. And so, because of this, we notice that all together, there are six lines that I’ve drawn there, so six possible pairings. And so here, the number of possibilities is 2 times 3.
The word and meant that we multiply the 2 and the 3. So this is a really big idea, a fundamental idea to keep in mind that or means add and and means multiply. Another important idea is listing the items you need to count. So for example, you have to make a bunch of arrangements, start listing sample arrangements.
Now, the test is gonna be extremely rare that you would be able to solve a counting question simply by listing all the possibilities. Often, as we’ll see, there are going to be hundreds, sometimes even thousands of possibilities. So it would be impossible to make a list in any kind of time efficient way. But listing possibilities is often a good first step that will give you insight into which approach to take.
So often, if you’re stuck in a counting problem, listing the first few possibilities is one way to start thinking about, gee, what method is gonna work best here? In summary, counting, or combinatorics, deals with strategies for counting large numbers of combinations and arrangements. The word and means multiply.
The word or means add. And remember that listing is a good preliminary strategy that may yield further insights.
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