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Question Video: Applications of the Counting Principle (Product Rule) Mathematics • 12th Grade

Use the fundamental counting principle to determine the total number of outcomes upon choosing from 8 ice cream flavors; small, medium, or large cones; and either caramel or chocolate sauce.

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Video Transcript

Use the fundamental counting principle to determine the total number of outcomes upon choosing from eight ice cream flavors; small, medium or large cones; and either caramel or chocolate sauce.

Lets begin by recalling what we mean by the fundamental counting principle. The fundamental counting principle, sometimes called the product rule for counting, tells us that the total number of outcomes for two or more events is found by multiplying the total number of outcomes for each event together. So what are the events were interested in? Well, event one is choosing an ice cream flavor. There are eight ice cream flavors, and so there are eight possible outcomes for event one.

Then we have event two, and thats the type of cone that we choose. Those are small, medium, or large, and so there are three possible outcomes for the type of cone we choose. Finally, we move onto event three. This is the flavor sauce that we end up choosing. We can choose between caramel and chocolate sauce, and so there are two possible outcomes for event three. The total number of outcomes, in other words, the total number of combinations of ice cream we can choose, is found by multiplying each of these values together. Thats eight times three times two.

And of course, since multiplication is commutative, we can do this in any order. And we can first begin by finding the product of three and two; thats six. So were actually calculating eight times six, which is, of course, 48. And so we see there are a total number of 48 outcomes of the type of ice cream cone we can choose.

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