Permutations
2026-02-28 10:45 Diff
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Last updated on November 25, 2025

Permutations refer to different ways of arranging objects in a specific order. It is the rearrangement of a set of items in a specific linear order. The symbol nPr is used to indicate the number of permutations of n distinct objects, taken r at a time. In this topic, let's learn about permutations in detail.

What are Permutations?

A permutation is an arrangement in which the order of the items matters.
Consider an example in which five chairs are available, and three people need to be seated.
 

  • There are five choices for the first chair.
     
  • After seating one person, four choices remain for the second chair.
     
  • Then, three choices are left for the third chair.
     

So, the total number of ways to arrange three people in 5 chairs is:

5 × 4 × 3 = 60 ways

Notice that this multiplication can be written using factorials:

\(5 \times 4 \times 3 = \frac{5!}{2!} = \frac{5!}{(5-3)!}\)

In general, when arranging r people in n chairs, the number of permutations is given by:

  \(nP_r = \frac{n!}{(n-r)!}\)

How to Calculate Permutations?

Here, let’s see how we can find permutations. The general formula we use to find permutations is:

P(n, r) = n! / (n − r)!

Here, n is the total number of elements in the data set.


r is the total number of selected elements in a specific order.


! is the factorial. 


For instance, let's say we have 10 different books and want to select and arrange 2 of them. We can calculate the number of ways to do it using permutations. 


Without repetition:


The formula is: nPr = 10! / (10 − 2)! = 10! / 8! = (10 × 9 × 8!) / 8!  = 90


So, there are 90 unique ways to arrange 2 books from a set of 10. 

With repetition, the formula is:


10\(^2\) = 10 × 10 = 100


Hence, there are 100 ways (with repetition) to arrange 2 books from a set of 10. 

Another key concept is factorials, and they are useful in permutations.

For example, the factorial of 8 = 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320.

Types of Permutation

Permutations has a different form depending on the rules for arranging the object. To understand how arrangements work in various situations, permutations are classified into three main types. Each type follows a different condition about repetition and the uniqueness of objects.

  • Permutation of n different objects (without repetition)
     
  • Permutation with repetition
     
  • Permutation of multisets (when some objects are identical)


1. Permutation Without Repetition

All objects are different, and no object is used more than once. Each time you choose an object, the number of choices decreases.

The formula is:

\(P(n, r) = n \times (n - 1) \times (n - 2) \times (n - r + 1)\)


This means we are arranging r objects from n distinct objects, and order matters.
 

2. Permutation With Repetition

Objects can be used again and again. Since repetition is allowed, the number of choices stays the same each time.

The formula is:

\(P_{\text{repetition}}(n, r) = n^r\)

This means each of the r positions can be filled in n ways.


3. Permutation of Multisets

In this type, some objects are repeated or identical. Since some items look the same, we adjust the count to avoid repeating the same arrangement.

A multiset has groups of identical items, and the formula ensures only unique arrangements are counted.
 

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Properties of Permutation

To understand permutations better, it helps to know a few important properties. These properties show how permutation values behave and how they can be simplified or related to each other.
 

Here are the commonly used properties of permutations:

  • \(nP_n = n \times (n-1) \times (n-2) \times \dots \times 1 = n!\)
  • \(nP_0 = \frac{n!}{(n-0)!} = \frac{n!}{n!} = 1\)
  • \(nP_1 = n\)
  • \(nP_{n-1} = \frac{n!}{(n-(n-1))!} = \frac{n!}{1!} = n!\)
  • \(\frac{nP_r}{nP_{r-1}} = n - r + 1\)
  • \(nP_r = n \cdot (n-1)P_{r-1} = n \cdot (n-1) \cdot (n-2)P_{r-2} = n \cdot (n-1) \cdot (n-2) \cdot (n-3)P_{r-3} \dots \), and so on.
     
  • \((n-1)P_r + r \times (n-1)P_{r-1} = nP_r\)

Difference Between Permutations and Combinations

Permutations and combinations are methods used to determine the number of possible arrangements of elements. So, let's see how they differ from each other. 

Permutation Combination In permutation, the order of the data is considered. In combinations, the order of data is not considered.  In permutations, elements are selected from a list. In combination, the data is chosen from a group. The data is specifically arranged. Here there is a selection of data.

Tips and Tricks for Permutation

Understanding permutations that become easier when the ideas are explained using the  simple language and real-life examples. These tips help make the concept clearer, more interactive, and easier to apply in different situations.
 

  • If you change the order, the arrangement becomes different.
     
  • Use small examples first to build understanding before solving larger problems.
     
  • Connect the concept to real-life examples, such as arranging books, toys, or snacks in different orders.
     
  • Use a permutation calculator when numbers become large to help children verify answers.
     
  • Explain permutation vs combination through simple scenarios: arranging students (permutation) vs choosing a group (combination).
     
  • Start with practical situations like arranging students in a line or organizing items on a shelf.
     
  • Simplify the process by multiplying just the first r terms rather than writing out the whole factorial.
     

Common Mistakes and How to Avoid Them in Permutations

Students tend to make mistakes when learning inter-related concepts like permutations and combinations. So let’s check out some common mistakes and ways to avoid them when learning permutation. 

Real-Life Applications of Permutations

Now let’s learn how we use permutations in the real world.

  • In cybersecurity and encryption, we use permutations to create passwords to secure our systems. 
  • In lottery games, we use permutations to pick from a pool of numbers. 
  • Permutations can be applied to various scenarios, such as arranging people, and seats, forming teams, and so on.
  • In DNA and genetics, permutations help in analyzing sequences of a DNA.
     
  • Permutations are used to organize schedules, plan tasks, or determine the order of events efficiently.
     
  • Permutations are used to determine the possible orders of winners, medal distributions, and match schedules.

Problem 1

In how many ways can 5 different books be arranged on a shelf?

Okay, lets begin

120
 

Explanation

 To find the permutations, we use the formula n! 


Here, n = 5


So n! \(= 5! = 5 × 4 × 3 × 2 × 1 = 120\)


So, we can arrange the books in 120 ways. 
 

Well explained 👍

Problem 2

How many 3-letter words (with distinct letters) can be formed using the letters A, B, C, D, and E?

Okay, lets begin

Here, we can arrange the given letters into 60 different three-letter words.
 

Explanation

To find the number of possibilities of 3-letter arrangements, we find the permutation using the formula: 


\(P(n, k) = \frac{n!}{(n - k)!}\)


Here, n = 5 and k = 3

\(P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!}\)


\(= \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1}\)


Next, cancel out the common terms (2 × 1):


So, \(5 × 4 × 3 = 60 \)

Well explained 👍

Problem 3

In how many ways can the letters in the word 'GOLD' be arranged?

Okay, lets begin

The word 'gold' can be arranged in 24 different ways.

Explanation

The word 'GOLD' can be arranged in n! ways.


Here, n = 4 (the word 'GOLD' consists of 4 letters).


So we start with 4 and then multiply 4 by the next smallest number, 3. 

\(4 × 3 = 12\)


Again, multiply the result by 2: 


\(12 × 2 = 24 \)


Finally, multiply 24 by 1. 


\(24 × 1 = 24. \)


So, \(n! = 4! = 4 × 3 × 2 × 1 = 24\)


So, the word 'GOLD' can be arranged in 24 different ways.
 

Well explained 👍

Problem 4

In how many ways can 8 athletes be assigned 3 distinct positions in a relay race?

Okay, lets begin

There are 336 ways for 8 athletes to try 3 distinct positions in a relay race.

Explanation

We use the permutation formula,

 
\(P(n, k) = \frac{n!}{(n - k)!}\)


Here, n = 8 and k = 3


So, \(P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!}\)


\(= \frac{8 \times 7 \times 6 \times 5!}{5!}\)


\(= 8 \times 7 \times 6 = 336\)
 

Well explained 👍

Problem 5

A locker password consists of 5 distinct digits chosen from 1 to 9. How many passwords can be created?

Okay, lets begin

15,120 passwords can be created.

Explanation

We use the permutation formula, 


\(P(n, k) = \frac{n!}{(n - k)!}\)


Here, n = 9 and k = 5


So, \(P(9, 5) = \frac{9!}{(9-5)!} = \frac{9!}{4!}\)


\(= \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4!}{4!}\)


\(= 9 × 8 × 7 × 6 × 5 = 15,120.\)
 

Well explained 👍

FAQs on Permutations

1.What is a permutation?

A permutation is the arrangement of objects in a specific order. Here, the order of selection matters. 
 

2.What is the formula for permutation?

3.What is the difference between a permutation and a combination?

The main difference between permutations and combinations is that in permutations the order matters but in combinations the order doesn't matter.

4.What are some real-life applications of permutations?

In real life, we use permutations to schedule events, organize shelves, create passwords, and so on.

5. What are the types of permutations?

The different types of permutations include permutations with repetition, without repetition, with multisets, and circular permutations. The two common types are permutations with repetition and without repetition. Permutations with multi-sets and circular permutations are less prevalent.
 

6.How can parents explain permutations to their child in a simple, understandable way?

Parents can describe permutations as different ways of arranging things, like how a child can rearrange toys or letters to form new orders.
 

7.How can parents help their child remember the permutation formula easily?

Parents can encourage the child to use short tricks, like focusing on the number of arrangements and practicing with small examples.
 

8.What can parents do if their child gets confused between permutations with repetition and without repetition?

Parents can show the child examples, such as repeating digits in a lock code (with repetition) or arranging unique objects (without repetition).

Jaipreet Kour Wazir

About the Author

Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref

Fun Fact

: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!