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Unformatted text preview: Chapter 6.3: The Rule of Product and Rule of Sum Today we will explore two basic techniques of counting. We will use these two rules in all of our subsequent work in counting. The Rule of Product: If there are x ways to perform a first task, and y ways to perform a second task, then there are xy ways to perform the first task followed by the second task. Example: I go to a restaurant for breakfast. Eggs can either be served fried or scrambled. My toast will either be made with white bread, whole- wheat bread, rye bread, or flax bread. How many different breakfasts are possible? The two tasks are: 1. selecting the way the eggs are prepared, and 2. selecting the type of bread. There are two ways to select the way the eggs are prepared. There are four ways to select the type of bread. By the rule of product, there are 2 4 = 8 different ways to select my eggs and toast. We can use a tree diagram to visualize the different types of break- fasts. In general, if a problem can be split into several different tasks to be done sequentially, then we can represent the problem with a tree diagram. Also, we can generalize the rule of product for situations where there are more than two tasks. In general, the number of ways to complete a sequence of tasks is equal to the product of the number of ways to complete each individual task. 1 Example: A coin is flipped three times. How many different outcomes are there? We can solve this problem with the rule of product. There are two different outcomes for the first flip (heads or tails), two for the second, and two for the third. All together, there are 2 2 2 = 8 different outcomes if a coin is flipped three times. We can also solve the problem by drawing a tree diagram. Since there are three different tasks (a first flip, a second flip, and a third flip) then the tree will branch out three times. Since each task has two different outcomes, then each time the tree branches out, it will split into two branches. Here is a tree diagram modeling this scenario: Note that on this tree diagram, there are eight branches. Each branch corresponds to one of the different outcomes. If we were to follow along each branch, we would have a list of each of the different outcomes....
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