Video: Application of the Counting Principle (Product Rule)

In a small town, there are 4 paths connecting the library and the post office, 5 paths connecting the post office and the bank, and 2 paths connecting the bank and the park. Determine the number of ways to walk from the library to the park.

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Video Transcript

In a small town, there are four paths connecting the library and the post office, five paths connecting the post office and the bank, and two paths connecting the bank and the park. Determine the number of ways to walk from the library to the park.

Now, let’s picture what’s going on. We’re first told that there are four possible paths that we can take to travel from the library to the post office. Then, there are five different paths that we can take to travel from the post office to the bank. And then, there are two possible paths that we can take to travel from the bank to the park. We’re looking to find the number of ways to walk from the library to the park.

And so, we begin by making an assumption. And that assumption is that there’s no direct path from the library to the park. We’re going to need to travel in the direction given, from the library to the post office, the post office to the bank, and then finally the bank to the park.

And so, to work out the total number of ways we can perform this journey, we need to take into account each individual set of paths. And so, we recall something called the fundamental counting principle or the product rule for counting. This tells us that we can find the total number of outcomes of two or more events combined by multiplying the number of outcomes for each event together.

And so, what are the events we’re interested in? Event one is traveling from the library to the post office. Event two is the second part of our journey. That’s traveling from the post office to the bank. And then, we have event three. That’s traveling from the bank to the park. We then see that event number one has four possible outcomes since we can take four different paths. There are five paths from the post office to the bank, and so event two has five outcomes. And similarly, event three has two possible outcomes.

The total number of ways that we can walk from the library to the park then is the product of these values. It’s four times five times two. Now, four times five is 20, and 20 times two is 40. There are 40 ways that we can walk from the library to the park.

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