Exercise 4 Page 311

We are asked how can a tree diagram help us visualize the number of ways in which two or more events can occur. Each branch on a tree diagram represents an outcome of an event. Because of this, the final branches of the diagram show the total number of ways that the outcomes could be matched. Let's illustrate this with an example!

Example

Consider that a person is going on vacation. The following list is the items found in their luggage.

• $3$ shirts
• $2$ bottoms — a pair of pants and a pair of shorts
• $2$ pairs of shoes — a pair of sneakers and a pair of dress shoes

Combining the items from this luggage in a variety of ways, they can make different outfits. Each potential outfit is one way in which the events of choosing items can occur. We can make a tree diagram to visualize these outfits. Let's begin with the shirts.

In the first level of the diagram, each branch represents one way in which we can choose one of the three shirts. Let's add the possible choices of bottoms to the tree diagram.

We can see that, after choosing a top, we can choose either the pair of pants or the pair of shorts. Each different branch of the tree diagram represents a selection of a top and a bottom. Finally, let's add the shoes!

We can see that, after choosing a top and a bottom, we can choose either pair of shoes. Following a path from beginning to end, we determine one way in which the three choices can occur. To find the total ways in which the three choices can occur, we will use the Fundamental Counting Principle. Looking at the diagram, we can see that each path represents one of the $12$ possible outcomes.