In this lesson, you will learn how to calculate the probability of a permutation by analyzing a real-world example in which the order of the events does matter. We’ll also review what a factorial is. We will then go over some examples for practice.

## Probability

A **probability** is the likelihood of an event occurring.

When calculating the probability of an event, use the formula number of favorable outcomes over the number of total outcomes. We can calculate the probability of any event occurring by examining the choices that are available. Let’s look at an example to see how simple it is to find the probability.

Jimmy bought a bag full of gumballs at the store. In the bag there were 20 red, 15 blue, and 12 yellow gumballs. If Jimmy reaches in the bag and selects a gumball at random, what is the probability that he will select a red gumball?The first thing that Jimmy knows is that if he adds the number of blue, red, and yellow gumballs together, he has a total of 47 gumballs in the bag. Jimmy knows that to calculate the probability of selecting a red gumball, he must use the equation *the number of favorable outcomes over the number of total outcomes*.Since there were 20 red gumballs, the number of favorable outcomes is 20. Also, there were 47 total gumballs, so the number of total outcomes is 47.

We can see that the probability of Jimmy selecting a red gumball from the bag is 20/47.To finish calculating the probability, we need to always check and see that our fraction is in its simplest form. Fortunately for Jimmy, his fraction is in simplest form. So the probability of Jimmy drawing a red gumball from the bag at random is 20/47.

## Permutations

To calculate the total number of outcomes, you occasionally have to use a **permutation**. A permutation is a method to calculate the number of events occurring where order matters. To calculate a permutation, you will need to use the formula *n*P*r* = *n*! / (*n* – *r*)!. In this equation, *n* represents the number of items to choose from and *r* represents how many items are being chosen.Another function of this equation is the use of the ! (exclamation point). This symbol stands for ‘factorial.

‘ A factorial is the product of all positive integers equal to and less than your number. A factorial is written as the number followed by an exclamation point. For example, to write the factorial of 5, you would write 5!.

To calculate the factorial of 5, you would multiply the positive integers equal to and less than 5. 5! = 5 x 4 x 3 x 2 x 1. By multiplying these numbers together, we can find that 5! = 120. Let’s look at another example: how would we write and solve the factorial of 10? The factorial of 10 would be written as 10!. To calculate: 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800.Let’s look at an example where we must calculate the permutation.

At Rockland University, it is time for the student government election. There are 20 students who have submitted the necessary paperwork to appear on the ballot. How many different ways can the student body select a President, Vice President, Secretary, and Treasurer?First of all, this example is a permutation because the order in which the students are selected for the positions does matter. We know that there are 20 students who have been chosen to run for office – this will represent the *n* variable in our equation. We also know that the student government consists of only 4 positions, which is our *r* term. So in this permutation, we have 20 items to choose from for the 4 positions. The equation for our permutation would look like 20P4 = 20! / (20 – 4)!.

To begin solving this permutation, we need to subtract the denominator. (20 – 4)! = 16! Next, let’s expand each factorial. As we work this equation, we’re going to cancel out common terms in both the numerator and denominator. We can see that there is a 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1 on both the top and bottom.

These terms can be cancelled out, leaving 20 x 19 x 18 x 17 on top and nothing on bottom, which becomes an understood 1.To solve this equation, we will now multiply the top and bottom of our fraction. 20 x 19 x 18 x 17 = 116,280. The denominator has a term of 1, so our fraction is 116,280/1. We can now see that there are 116,280 ways that the student government can be selected from those 20 students.

## Calculating Probability of Permutations

When calculating the probability of a permutation, we will be combining these two skills.

The use of permutations will give us the total number of outcomes and then we will need to calculate the total number of favorable outcomes. Let’s look at an example to see how these two skills can be combined.Jim was given a new smartphone with a 5-digit passcode. The passcode could contain any single digit number (0-9), but each number cannot be repeated. Unfortunately, Jim has forgotten his passcode and must try to guess the number. What is the probability of Jim guessing the correct 5-digit passcode?This problem will involve two processes: first of all, the use of permutations to calculate the number of passcodes that Jim could choose from, and secondly, the probabilities to calculate the chances of Jim selecting the correct passcode.

First, we need to calculate the permutation of Jim’s total outcomes. There are 10 items, and 5 of them are being chosen. This permutation would look like 10P5 when written in notation form. Next, we will need to plug in our values into the formula for a permutation. Remember, the equation to calculate a permutation is *n*P*r* = *n*! / (*n* – *r*)!.

Let’s plug in our values. For all of the *n* variables we will plug in a 10 and for the *r* variables a 5. Our equation now looks like 10P5 = 10! / (10 – 5)!. To work this equation, we need to subtract on the bottom: (10 – 5) = 5!.

We now have 10!/5!. Next, let’s expand our factorials. On top, 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1, and on bottom, 5! = 5 x 4 x 3 x 2 x 1.

As we continue working this equation, we will cancel out common terms in both the numerator and denominator. Since there is 5 x 4 x 3 x 2 x 1 on both the top and bottom, we can cancel out these terms. As these terms are canceled out, they disappear. Our equation now looks like 10 x 9 x 8 x 7 x 6 /1. As we multiply, 10 x 9 x 8 x 7 x 6 = 30,240. This means that there are 30,240 different outcomes for passcodes that Jim could use.To calculate the probability that Jim selects the correct passcode, we will use the equation *total number of favorable outcomes over the total number of outcomes*.

Since there is only one possible passcode that will unlock the phone, the number of favorable outcomes is 1. After calculating the permutation of events, we know that there are 30,240 total outcomes. So the probability that Jim selects the correct passcode is 1/30,240.

## Lesson Summary

In this video, we’ve looked at several examples in which we calculate the probability of a permutation. To find the probability of an event occurring, we will use the equation *number of favorable outcomes over the number of total outcomes*.When calculating the total number of outcomes, we sometimes need to use a permutation. A **permutation** is a method to calculate the number of events occurring where order matters. To calculate a permutation, we will need to use the formula *n*P*r* = *n*! / (*n* – *r*)!. In this equation, *n* represents the number of items to choose from and *r* represents how many items are being chosen.

Once you have found the permutation, you will plug it in as the total number of outcomes. Remember after you find the probability that all probabilities must be in simplest form.

## Learning Outcome

After viewing the lesson, you should be able to:

- Define what permutations are
- Solve for probability
- Calculate a probability problem using permutations