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 more difficult examples.
It’s dangerous to try to come up with a simple guideline for when to use the product rule and when to use the sum rule, because such a guideline will often go wrong in complicated situations. Nonetheless, a good question to ask yourself when you are trying to decide which rule to use is, “Would I describe this with the word ‘and,’ or the word ‘or’?” The word “and” is generally used in situations where it’s appropriate to use the product rule, while “or” tends to go along with the sum rule.
Let’s see how this applies to each of the examples we’ve looked at in this chapter.
In Example 2.1.1, you needed to choose the size and the variety for your coffee. In Example 2.1.2, Kyle wanted to choose a doughnut and coffee. In Example 2.1.3, Chlöe needed to determine the size and the colour and the image and the slogan for each t-shirt. In Example 2.1.4, we wanted to know the sex of Peter and Mary’s third and fourth children. So in each of these examples, we used the product rule.
In Example 2.2.1, you needed to choose a bagel or a doughnut. In Example 2.2.2, Mary and Peter could have zero or one or two or three children. So in each of these examples, we used the sum rule.
You definitely have to be careful in applying this guideline, as problems can be phrased in a misleading way. We could have said that in Example 2.2.1, we want to know how many different kinds of doughnuts and of bagels there are, altogether. The important point is that you aren’t choosing both of these things, though; you are choosing just one thing, and it will be either a doughnut, or a bagel.
In Example 2.3.1, Grace was choosing a main dish and a side dish and a beverage, so we used the product rule to put these aspects together. Whether or not she had extra options available for her main dish depended on whether she chose pancakes or waffles or oatmeal or omelette or nothing, so the sum rule applied here. (Note that we didn’t actually consider each of these four things separately, since they naturally fell into two categories. However, we would have reached the same answer if we had considered each of them separately.) Similarly, whether or not she had extra options available for her side dish depended on whether she chose toast or not, so again the sum rule applied.
In Example 2.3.2, the plates can be regular (in either of two ways) or veteran or motorcycle plates, so the sum rule was used. In each of these categories, we had to consider the options for the first character and the second character (and so on), so the product rule applied.
Finally, in Example 2.3.3, the shirt Kathy chooses can be short-sleeved or long-sleeved, so the sum rule applies to that distinction. Since she wants to choose a shirt and gift wrap, the product rule applies to that combination.
For each of the following problems, do you need to use the sum rule, the product rule, or both?
- Count all of the numbers that have exactly two digits, and the numbers that have exactly four digits.
- How many possible outcomes are there from rolling a red die and a yellow die?
- How many possible outcomes are there from rolling three dice, if you only count the outcomes that involve at most one of the dice coming up as a one?