Permutations (Definition, types of permutations, and applications!)

A permutation is defined as the arrangement of a set of objects in a particular order. While this definition may sound simple, permutation is much more complicated than it appears on the surface.

So, what is permutation? The word permutation describes a mathematical calculation of the number of ways a specific set can be arranged. In other words, a permutation represents the number of ways something can be ordered or arranged. Most importantly, with permutations, the order of the arrangement matters.

In this post, we will take a closer look at what permutation is, how it is used in mathematics, and some of its applications in everyday life. Check out this article if you wonder what are the differences between permutation and combination.

Understanding Permutation

Permutation is a mathematical technique used to count the number of possible arrangements of objects in a given order. Generally, permutation is used in different fields, including computer science, statistics, combinatorics, and physics.

Below is the formula for calculating permutation: kn= n! /(n-k)! where n is the total number of objects and k is the number of objects taken at a time.

Permutation Formula
Permutation Formula

For instance, to calculate the total possible arrangement of 3 objects taken at a time from 6 different objects, we would first calculate 6!/(6-3)! which is equal to 654=120.

Types of Permutations

Permutations can be classified into 4 types permutations:

  • Permutations with repetitions
  • Permutations without repetitions
  • Circular permutations
  • Derangements

1- Permutations with Repetitions

Permutations with repetitions are commonly used in cases where there is a repetition of items. In other words, the order of items matters even if they are repeated. For instance, suppose one wants to arrange the letters in the word “ABA,” there are three possible arrangements (AAB, ABA, BAA), even though there are only two letters.

Keep in mind that permutations with repetitions can be challenging to calculate as there can be many repetitions, and the formula must account for all possible permutations.

Check out the video below to learn more about permutations with repetitions.

Permutations with Repetitions

2- Permutations without Repetitions

Permutations without repetitions are the most common type of permutation that most of us are familiar with. Here, the order of elements matters, and there are no repetitions.

For instance, if we have six different books to arrange on the shelf, there are 720 different ways in which we can arrange them. The formula for calculating permutations without repetition is n! / (n-r)! where n is the number of elements and r is the size of the group.

I encourage you to watch the video below to learn more about permutations without repetitions.

Permutations without Repetitions

3- Circular Permutations

A circular permutation is where the arrangement of items on a circle matters. Items on a circle can be rotated to any degree without changing the order.

For instance, if six people have to be arranged in a circle, and there is no specific position for each person, there are five possible arrangements. The formula for calculating circular permutations is Pn = (n-1)!

The video below discusses circular permutations.

Circular Permutations

4- Derangements

Derangement is a special type of permutation where none of the elements retains its original position. In other words, if a group of items is randomly chosen and arranged, the probability that no element is in its original position is a derangement.

The number of derangements for a group of n elements is given by the formula n! (1/0! – 1/1! + 1/2! – 1/3! +…+ ((-1)n)/n!).

I invite you to check out Illinois State University or watch the video below to learn more about derangements.


Uses of Permutation in Mathematics

Permutation is a fundamental concept in mathematics and is used extensively in various areas of the field. For instance, it is used to solve probability problems, counting principles, and coding theory.

In probability, permutations are used to find the number of outcomes that are possible given a specific scenario. To calculate the probability of a specific event happening, we need to calculate the total possible number of outcomes and the total number of favorable outcomes.

The same technique is used in coding theory, where a specific arrangement of symbols may represent different letters or numbers.

I encourage you to head over to Khan Academy or watch this video to learn more about permutations.

Permutation in Everyday Life

Permutations also have a wide range of industrial applications, including inventory management, supply chain optimization, and workforce scheduling.

  • Anagrams are a perfect example of how permutation can be used in everyday life.
  • Permutations are also useful in generating unique passwords, license plate numbers, and lottery numbers.
  • Permutations are widely used in computer science, particularly in cryptography and coding.
    • Cryptography uses permutations to encode and protect data, such as passwords and information security.
    • Coding uses permutations in algorithms to achieve various functionalities, such as sorting and searching data.
    • Permutation functions are also used in data analysis, particularly in statistics, where it is used to analyze and understand the relationship between different data sets.
  • Aside from its applications in computer science, permutation also plays a significant role in other sciences, such as chemistry and astronomy.
    • In chemistry, permutations are crucial to understanding molecules’ structure and behavior at the molecular level.
    • In astronomy, permutation is used to predict the motion and position of celestial objects, such as planets and stars.
  • In inventory management, permutations are used to ensure that there is enough stock on hand to meet the demand.
  • In supply chain optimization, it’s essential to consider the order in which products are arranged to minimize the cost of shipping and handling.
  • In workforce scheduling, the ability to rearrange teams leads to higher productivity and efficiency.

What to read next:

Wrapping Up

A permutation is a vital concept in mathematics and has various applications in everyday life and industry. A permutation is all about arranging objects, symbols, or people in a particular order and determining the possible number of outcomes.

Also, permutations play a significant role in many areas, including statistics, probability, and cryptography.

I believe that by understanding the four different types of permutations: permutations with repetition, permutations without repetition, circular permutations, and derangements, you will find it easier to solve complex problems that involve sorting and counting.


I am Altiné. I am the guy behind When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting

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