Permutation: Definition, Formula & Examples

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  • 0:02 Introduction to Permutations
  • 0:17 Involving Repetitions
  • 1:19 Involving No Repetitions
  • 3:10 Any Sized Group of Objects
  • 4:31 Examples
  • 6:15 Lesson Summary
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Lesson Transcript
Norair Sarkissian

Norair holds master's degrees in electrical engineering and mathematics

Expert Contributor
Jerry Allison

Jerry holds a Doctor of Business Administration and a Master’s in Mathematics. He has taught business, math, and accounting for over 25 years.

In this lesson, we will examine a mathematical method of calculating the number of ways in which we can order a group of distinct objects. This involves the use of a mathematical concept known as a permutation.

Introduction to Permutations

A permutation is an arrangement or ordering of a number of distinct objects. For example, the words 'top' and 'pot' represent two different permutations (or arrangements) of the same three letters.

Permutations Involving Repetitions

Let's say that we are taking a test, which consists of five multiple choice questions. There are four different answers (a through d) to choose from for each question. What is the total number of ways in which we could answer the five questions on the test?

One way of thinking about this problem is as follows: let's focus on just the first question on the test. How many choices do we have? The answer to this question is four, since we can pick any answer from a to d. Then we ask the same question for each successive question, and clearly, there are four ways of answering each of them. Therefore, the total number of ways for filling out the test is the product of:


Since we have the same number of choices for each question, the same scenario is repeated for each question on the test. We can generalize our calculations and state that if there are n choices and r repetitions, then the number of permutations or arrangements is given by:


Permutations Involving No Repetitions

Now let's consider a somewhat different situation. Let's assume that we have four cell phones of the same make and model, but each one is a different color, so they are distinct objects. We buy four distinct cases for them. How many ways are there for us to place the four different cell phones in the four cases?

We can list all the possible arrangements and then count them:


We can see that there are 24 unique permutations. We would like to know if it is possible to calculate the total number of permutations without listing all the arrangements and then counting them. It turns out that it's pretty easy to do this if we use a systematic approach to the problem.

Let's start with the first cell phone case: how many choices do we have? The answer is four, since any of the four cell phones could be placed in the first case. When we ask the same question about the second case, we realize that we no longer have four choices, because we already placed a cell phone in the first case, and we now have three cell phones remaining. This means that we have three choices when placing a phone in the second case.

As for the third case, we only have two phones remaining, or two options left. And when we get to the fourth case, we only have a single phone to place in it, so there is only one choice. Thus, the number of ways to place four different cell phones in four different cases is (4)(3)(2)(1) = 24.

Comparing this scenario with the previous one involving the multiple choice test, we can see that there are no repetitions here, since once an object is selected (such as a cell phone placed in a case), it's no longer available for selection. When no repetitions are involved, the number of permutations of n distinct objects is given by n!, where n! is the factorial of the integer n.

Any Sized Groups Of Objects

This scenario involves no repetitions, but it's different from the previous situation. For example, starting with the same four cell phones of different colors, how many arrangements of two phones can we form? In mathematical language, we describe this situation as the number of permutations of 4 objects, taken two at a time.

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Additional Activities

Permutation Thought Questions


You will need a piece of paper, a pencil, and a penny, a nickel, and a dime. Take the piece of paper and draw three circles, labeling one of the circles as #1, another of the circles as #2, and the final circle as #3. Place the coins into the circles on the paper and record which coin is in circle #1, which is in circle #2, and which is in circle #3. Now find another arrangement and write it down. Continue this process until all the possibilities have been found. How many ways of doing this did you find? Is this a permutation with repetitions or not? Which formula would you use to calculate the total number of arrangements? Calculate the number of arrangements. Does your experiment number agree with the calculated number?

Calculation Problem

Electra Cute Company designs signal transmitters that transmit binary digits (1s and 0s) also known as bits. The company wishes to test the transmitter by sending all possible 16-bit strings consisting of a mixture of 1s and 0s. Is this a permutation with or without repetitions? Which formula would be used? How many possible 16-bit strings are possible? How would you feel about listing all of them?

Research Project

Use the Internet to find out how many horses ran in the Kentucky Derby last time. We would like to calculate how many first, second, and third place finishes are possible (we do not care about the horses that finish after third). Is this a permutation with or without repetitions? What formula would you use? Do the calculation and find the answer. Now suppose the Kentucky Derby decides to add one more horse to the race next year. How does that affect the calculation? What is your new answer for the number of first, second, and third place finishes?

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