2: Basic Counting Techniques

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Joy Morris
University of Lethbridge

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When we are trying to count the number of ways in which something can happen, sometimes the answer is very obvious. For example, if a doughnut shop has five different kinds of doughnuts for sale and you are planning to buy one doughnut, then you have five choices.

There are some ways in which the situation can become slightly more complicated. For example, perhaps you haven’t decided whether you’ll buy a doughnut or a bagel, and the store also sells three kinds of bagels. Or perhaps you want a cup of coffee to go with your doughnut, and there are four different kinds of coffee, each of which comes in three different sizes.

These particular examples are fairly small and straightforward, and you could list all of the possible options if you wished. The goal of this chapter is to use simple examples like these to demonstrate two rules that allow us to count the outcomes not only in these situations, but in much more complicated circumstances. These rules are the product rule and the sum rule.

2.1: The Product Rule
The product rule is a rule that applies when we there is more than one variable (i.e. thing that can change) involved in determining the final outcome.
2.2: The Sum Rule
The sum rule is a rule that can be applied to determine the number of possible outcomes when there are two different things that you might choose to do (and various ways in which you can do each of them), and you cannot do both of them. Often, it is applied when there is a natural way of breaking the outcomes down into cases.
2.3: Putting Them Together
When we combine the product rule and the sum rule, we can explore more challenging questions. It doesn’t always happen that the sum rule is applied first to break the problem down into cases, followed by the product rule within each case. In some problems, these might occur in the other order. Sometimes there may seem to be one “obvious” way to look at the problem, but often there is more than one equally effective analysis, and different analyses might begin with different rules.
2.4: Summing Up
Very likely you’ve used the sum rule or the product rule when counting simple things, without even stopping to think about what you were doing. The reason we’re going through each of them very slowly and carefully, is because when we start looking at more complicated problems, our uses of the sum and product rules will become more subtle. If we don’t have a very clear understanding in very simple situations of what we are doing and why, we’ll be completely lost when we get to difficult examples.
2.5: Summary
This page contains the summary of the topics covered in Chapter 2.

Thumbnail: The abacus is a calculating tool that has been in use since ancient times and is still in use today. The abacus consists of a number of rows of movable beads or other objects, which represent digits. One of two numbers is set up, and the beads are manipulated to implement an operation involving a second number (e.g., addition), or rarely a square or cubic root. (Unsplash Lisense; Crissy Jarvis via Unspash)