**GMAT Exam Review Fundamental Counting Principle**

This video will introduce you to the Fundamental Counting Principle and how it’s used on the GMAT Test.

## GMAT Fundamental Counting Principle Review [Transcript]

In this lesson we will examine the Fundamental Counting Principle, which is the foundation of one of the most important strategies for tackling counting questions. We will examine the Fundamental Counting Principle through an example. In our example we are building a car that has several options. First, we can choose the car’s power source. The car can be gas-powered, electric, or hybrid. Next we can choose the car’s color. The car can be red, black, or blue. Finally we have two options for the car’s transmission, automatic or standard. So given all these options how many different cars can we build? To answer this we’ll use a tree diagram. We’ll take the task of building a car and break it into stages. One stage will be selecting the car’s power source. Another stage will be selecting the car’s color, and the last stage will be selecting the transmission.

Let’s begin by selecting the car’s power source. Here we have three options. Now if we choose to have a gas-powered car, then we have three options for the color of that car. Similarly if we choose to have an electric car, then we have three options for the color, and so on for the hybrid. Now if we choose to have a gas-powered car that is red, then we have two options for the transmission. Similarly if we choose to have a gas-powered car that is black, we have two options for the transmission. Now this process continues until we have accomplished all three stages. Now that our tree is complete, notice that all of the different car configurations can be found in the various paths that exist in our tree.

For example, this path represents a blue gas-powered car with an automatic transmission, this path represents a red electric-powered car with a standard transmission, and this path represents a hybrid car that is black with an automatic transmission. So the total number of different car configurations will be equal to the total number of different paths that exist in this tree. So how many paths are there? Well the total number of different paths will equal the total number of different endpoints, or leaves, on the tree. Notice that for each leaf there is a unique path.

For example, this leaf is the last stage of this path, representing a blue gas-powered car with automatic transmission. So since this tree has 18 leaves, there are 18 possible paths. Which means that it is possible to create 18 different cars given the following options. Okay, here comes the most important part. Notice that when we were constructing our tree diagram, we had three options for the power source, three options for the color, and two options for the transmission. Notice that when we find the product of these three values we get 18, the same number we calculated for the total number of different cars we can build. This is no coincidence. In fact, this result complies directly with the Fundamental Counting Principle.

Here’s how the Fundamental Counting Principle works. If we have some task that is made up of stages, where one stage can be accomplished in A ways, another stage in B ways, another stage in C ways and so on, then the total number of ways to accomplish the entire task will be equal to the product A times B times C and so on. So let’s apply this principle to our question. Here we can take the task of building a car and break it into three stages. One stage is selecting the car’s power source, another stage is selecting the car’s color, and another stage is selecting the car’s transmission. Let’s begin with the first stage. In how many ways can we select the power source? Well we can accomplish this stage in three ways. In how many ways can we choose a color? Well we can accomplish this stage in three ways as well. And finally there are two ways in which we can accomplish the last stage. So applying the Fundamental Counting Principle, the total number of ways to accomplish all three stages and build our car will be equal to the product of the number of ways to accomplish each stage. This product equals 18, so we can create 18 different cars.

Okay, that’s how the Fundamental Counting Principle works. Remember that at least half of all counting questions can be solved using this principle. So when you encounter a counting question, be sure to ask yourself “can I take the task of building possible outcomes and break it into individual stages?” If so, you may be able to apply the Fundamental Counting Principle.

**GMAT Test Prep**.