Permutation: Definition, Formula, Types, and Examples

Edited By Komal Miglani | Updated on Jul 02, 2025 07:43 PM IST

Permutation basically means the arrangement of things. And when we talk about arrangement then the order becomes important if the things to be arranged are different from each other (when things to be arranged are the same then order doesn’t have any role to play). So in permutations order of objects becomes important. In real life, we use permutation for arranging numbers, letters, codes, and alphabets.

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Permutation: Definition
Permutation Formula
Arranging n objects in r places (Same as arranging n objects taken r at a time) is equivalent to filling r places from n things.
Permutation of n different objects
Solved Examples Based on Permutation
Example 1: Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is[JEE MAINS 2023]
The permutations represent the number of distinct arrangements of objects where the order matters. The calculation of permutations using factorial notation provides a precise method to quantify and analyze sequential arrangements. Understanding of permutation is necessary for solving complex problems.

Permutation: Definition, Formula, Types, and Examples

In this article, we will cover the Introduction Of Permutation. This topic falls under the broader category of Permutations and combinations, which is a crucial chapter in Class 11 Mathematics. This is very important not only for board exams but also for competitive exams, which even include the Joint Entrance Examination Main and other entrance exams: SRM Joint Engineering Entrance, BITSAT, WBJEE, and BCECE. A total of thirteen questions have been asked on this topic in JEE Main from 2013 to 2023 including one in 2013, one in 2014, two in 2020, one in 2021, and eight in 2023.

Permutation: Definition

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement.

Arranging n objects in r places (Same as arranging n objects taken r at a time) is equivalent to filling $r$ places from $n$ things.

For example, suppose we have a set of three letters: A, B, and C. We want to find the number of ways in which 2 letters from this set can be arranged. Each possible arrangement would be an example of a permutation. The complete list of possible permutations would be $A B, A C$ , $B A, B C, C A$ , and $C B$ . Thus there 6 number of permutations. We observe that the order in which letters are occurring is important, i.e., AB and BA are two different arrangements. In mathematics, we use a specific terminology. That is "permutations as $n$ distinct objects taken $r$ at a time". Here n refers to the number of objects from which the permutation is formed, and $r$ refers to the number of objects used to form the permutation. In the above example, the permutation was formed from 3 letters (A, B, and C), so $n = 3$ and the permutation consisted of 2 letters, so $r =$ .

Permutation Formula

Arranging $n$ objects in $r$ places (Same as arranging $n$ objects taken $r$ at a time) is equivalent to filling $r$ places from $n$ things.

r places:
$Unknown environment 'tabular'$
$◻$ number of choices: $n$ ( $n - 1) (n - 2) (n - 3)$
$n - (r - 1)$

So the number of ways of arranging $n$ objects taken $r$ at a time $= n (n - 1)$ $(n - 2) \dots (n - r + 1)$

$\frac{n (n - 1) (n - 2) \dots (n - r + 1) (n - r)!}{(n - r)!} = \frac{n!}{(n - r)!} =^{n} P_{r}$

Where $r \leq n$ and $r \in W$

So, the number of ways arranging n different objects taken all at a time $=$ $^{ı} P_{n} = n!$

Example: In how many ways can 5 people be seated at 3 places?

Solution: Basically this question is about arranging 5 people at 3 different places

Let's think that we are given 3 places, so for the first place we have 5 people to choose from, hence this can be done in 5 ways as all 5 are available.

Now for 2nd place we have 4 people to choose from, hence this can be done in 4 ways.

Similarly, for 3rd place, we have 3 choices.
Since we have to choose for all 3 places, so multiplication rule is applicable, and the total number of ways $5 \times 4 \times 3 = 120$ ,

This can also be done directly from the notation or formula

$^{n} P_{r} where n = 5, r = 3, so^{5} P_{3} = \frac{5!}{2!} = 5 \times 4 \times 3 = 120$
Example: Find the number of ways the letters of the word "BIRTHDAY" can be arranged taken all at a time.

Solution: From the above concept directly using the formula $^{n} P_{n}$ we have

$^{8} P_{8} = 8! = 40, 320$

Types of Permutation

Permutation can be classified into three different categories:

Permutation of n different objects (when repetition is not allowed)
Repetition, where repetition is allowed
Permutation when the objects are not distinct (Permutation of multisets)

Permutation of n different objects

If $n$ is a positive integer and $r$ is a whole number, such that $r < n$ , then $P (n$ , r) represents the number of all possible arrangements or permutations of n distinct objects taken r at a time. In the case of permutation without repetition, the number of available choices will be reduced each time. It can also be represented as $^{n} P_{r}$ .

$P (n, r) = n! / (n - r)!$

Permutation when repetition is allowed

When the number of objects is " $n$ ," and we have " $r$ " to be the selection of object, then;

Choosing an object can be in $n$ different ways.
Thus, the permutation of objects when repetition is allowed will be equal to,

$n \times n \times n \times \dots \dots (r times) = n^{r}$
Permutation of multi-sets

Permutation of n different objects when $P_{1}$ objects among ' $n$ ' objects are similar, $P_{2}$ objects of the second kind are similar, $P_{3}$ objects of the third kind are similar and so on, $P_{k}$ objects of the kth kind are similar and the remaining of all is of a different kind,

Thus it forms a multiset, where the permutation is given as:

$\frac{n!}{P_{1}! P_{2}! \dots P_{n}!}$

Relation Between Combination And Permutations

The relation between combinations and permutations is given by

$^{n} C_{r} \times r! =^{n} P_{r}$
Factorial notation

Many times we multiplied consecutive integers. On the basis of this factorial notation is devised. In the counting principle many times answer is written in the form of factorial to save us time. The product of first $n$ natural numbers is denoted by $n!$ and is read as 'factorial n '.

Note:
The factorial of zero is defined and its value is one.

Solved Examples Based on Permutation

Example 1: Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is
[JEE MAINS 2023]

Solution

4 3 0 0 0 0 0 0 0 0999 V2000
6.2033 -1.3790 0.0000 R 0 0 0 0 0 0 0 0 0 0 0 0
2.0327 -5.6822 0.0000 R 0 0 0 0 0 0 0 0 0 0 0 0
7.5350 -5.2274 0.0000 R 0 0 0 0 0 0 0 0 0 0 0 0
6.3785 -9.2507 0.0000 R 0 0 0 0 0 0 0 0 0 0 0 0
1 2 1 0
2 3 1 0
2 4 1 0
A 1
3
A 2
8
A 3
3
A 4
2
M END
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$\begin{aligned} Ways = \frac{8!}{3! 3! 2! 2!} \times 3! \\ = \frac{8 \times 7 \times 6 \times 5 \times 4}{4} \\ = 56 \times 30 \\ = 1680 \end{aligned}$

Hence, the answer is 1680

Example 2: The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is
[JEE MAINS 2023]

Solution

G H O TU
$Unknown environment 'tabular'$ & & \
\hline & TOU & & & \
\hline & TOUG \lfloor & & 1 & \
\hline & & & 89 & \
\hline
\end{tabular}

Hence, the answer is 89

Example 3: The number of seven digits odd numbers, that can be formed using all the seven digits $1, 2, 2, 2, 3, 3, 5$ is
[JEE MAINS 2023]
Solution The no. of 7-digit odd Numbers that can be formed using 1, 2, 2, $2, 3, 3, 5$

$Unknown environment 'tabular'$
$\frac{\underset{―}{6}}{\underset{―}{3} ⌊ 2} = \frac{720}{12} = 60$ $= 240$

Hence, the answer is 240
Example 4: The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is
[JEE MAINS 2023]
Solution: UNIVERSE:
$Unknown environment 'tabular'$

= 240

Hence, the answer is 240

Example 4: The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is [JEE MAINS 2023]

Solution: UNIVERSE:

Vowels	Consonant
E, E	N, V
I, U	R, S

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2 vowels different, 2 consonants different

$\begin{aligned} (^{3} C_{2}) (^{4} C_{2}) (4!) \\ = (3) (6) (24) \\ = 432 \end{aligned}$

Hence, the answer is 432.

Example 5: The number of integers, greater than 7000 that can be formed, using the digit $3, 5, 6, 7, 8$ without repetition, is
[JEE MAINS 2023]
Solution
C-1

$\begin{aligned} 2 \times 4 \times 3 \times 2 = 48 \\ C - 2 5! (5 digit nos) = \frac{120}{168} \end{aligned}$

Hence, the answer is 168

Summary

The permutations represent the number of distinct arrangements of objects where the order matters. The calculation of permutations using factorial notation provides a precise method to quantify and analyze sequential arrangements. Understanding of permutation is necessary for solving complex problems.

Frequently Asked Questions (FAQs)

1. What is the difference between nPn and nPr?

nPn represents permutations of all n objects, which is simply n!. nPr represents permutations of r objects chosen from n objects, where r can be less than or equal to n. nPr is calculated as n! / (n-r)!

2. How do you calculate partial permutations?

Partial permutations involve selecting and arranging a subset of objects from a larger set. They are calculated using the formula nPr = n! / (n-r)!, where n is the total number of objects and r is the number being arranged.

3. How do you calculate permutations with repetition?

For permutations with repetition, if you have n types of objects and are making arrangements of length r, the number of permutations is n^r. For example, if you have 3 colors and want to make 4-digit codes, you have 3^4 = 81 possible permutations.

4. What is the permutation of n objects with repetition allowed?

When repetition is allowed in permutations of n objects, each position can be filled by any of the n objects. Therefore, the total number of permutations is n^r, where r is the number of positions to be filled.

5. How do permutations apply to probability problems?

Permutations are often used in probability to calculate the number of possible outcomes in an event where order matters. For example, when calculating the probability of correctly guessing a 4-digit PIN, we use permutations to determine the total number of possible PINs.

6. What does nPr represent in permutation notation?

nPr represents the number of permutations of n distinct objects taken r at a time. It's read as "n permute r" and calculated using the formula: nPr = n! / (n-r)!

7. How do you calculate 5P3?

To calculate 5P3, we use the formula nPr = n! / (n-r)!

8. What is the permutation of the word "MATH"?

Since "MATH" has 4 distinct letters, the number of permutations is 4! = 4 × 3 × 2 × 1 = 24. This means there are 24 different ways to arrange the letters in "MATH".

9. What is the connection between permutations and factorials?

Factorials are fundamental to permutations. The number of permutations of n distinct objects is n!, and factorials appear in various permutation formulas. Understanding factorials is crucial for mastering permutations.

10. How do you handle permutations with identical objects?

When dealing with permutations that include identical objects, we divide the total number of permutations by the factorial of the number of each identical object. For example, the permutations of the word "MISSISSIPPI" would be 11! / (4! × 4! × 2!).

11. What is the formula for calculating permutations of n distinct objects?

The formula for permutations of n distinct objects is n! (n factorial). For example, the number of ways to arrange 5 distinct objects is 5! = 5 × 4 × 3 × 2 × 1 = 120.

12. What is the difference between permutations with and without repetition?

Permutations without repetition involve arranging distinct objects, where each object can be used only once. Permutations with repetition allow objects to be used multiple times. The formulas and calculations differ for these two types.

13. What is a circular permutation?

A circular permutation is an arrangement of objects in a circle where rotations are considered the same permutation. For n objects, the number of circular permutations is (n-1)!

14. How many circular permutations are possible with 5 people sitting around a table?

For 5 people sitting around a table, we use the circular permutation formula: (n-1)!

15. What is a derangement in permutations?

A derangement is a permutation where no element appears in its original position. For example, in the set {1,2,3}, the permutation {2,3,1} is a derangement because no number is in its original position.

16. What is a permutation in mathematics?

A permutation is an arrangement of objects in a specific order. It represents the number of ways to organize a set of items where the order matters. For example, arranging 3 books on a shelf can be done in 6 different ways, and each arrangement is a permutation.

17. How does a permutation differ from a combination?

The key difference is that in permutations, the order of selection matters, while in combinations, it doesn't. For instance, when selecting a president and vice president from a group, the order matters (permutation). But when choosing two team members from a group, the order doesn't matter (combination).

18. What is the difference between permutations and arrangements?

In mathematics, "permutations" and "arrangements" are often used interchangeably. Both refer to the ordering of objects. However, "arrangement" is sometimes used more broadly to include combinations as well, while "permutation" specifically refers to ordered arrangements.

19. What is the difference between permutations and variations?

In some contexts, particularly in non-English speaking countries, "variations" is used to mean the same thing as permutations. Both terms refer to arrangements where order matters. However, "permutations" is the more commonly used term in English-speaking mathematics.

20. What is the role of permutations in combinatorics?

Permutations are a fundamental concept in combinatorics, the branch of mathematics dealing with counting. They are used to solve various counting problems, especially those where the order of selection matters, and form the basis for more complex combinatorial concepts.

21. What is the role of permutations in coding theory and error-correcting codes?

Permutations are used in various aspects of coding theory. They're employed in interleaving techniques to spread burst errors, in the construction of certain types of codes, and in analyzing the properties of code words. Understanding permutations helps in designing efficient and robust error-correcting codes.

22. How do permutations apply to the study of Cayley graphs?

Cayley graphs are visual representations of groups, often permutation groups. Each vertex represents a group element (a permutation), and edges represent the action of generators. Studying these graphs helps visualize group structure and properties, connecting permutations to graph theory and abstract algebra.

23. How do you solve a problem involving permutations with constraints?

To solve permutation problems with constraints, first identify the total number of objects and any restrictions. Then, break the problem into steps, applying the appropriate permutation formula for each step. Finally, multiply the results of each step together.

24. How do permutations relate to the Polya enumeration theorem?

Polya's enumeration theorem uses permutations and group theory to count the number of distinct colorings of a set of objects under certain symmetry conditions. It's a powerful tool in combinatorics that applies permutation concepts to solve complex counting problems in chemistry, computer science, and other fields.

25. How do permutations apply to the concept of derangements in probability?

Derangements, which are permutations where no element remains in its original position, are important in probability theory. They're used to solve problems like the "hat check" problem or calculating the probability of no matches in random assignments. The number of derangements is closely related to e, the base of natural logarithms.

26. How do you calculate the number of derangements?

The number of derangements for n objects is given by the nearest integer to n!/e, where e is the mathematical constant (approximately 2.71828). This can also be calculated recursively using the formula: D(n) = (n-1)[D(n-1) + D(n-2)], where D(0) = 1 and D(1) = 0.

27. How do you solve permutation problems involving "at least" or "at most" conditions?

For "at least" conditions, calculate the total permutations minus the permutations that don't meet the condition. For "at most" conditions, sum the permutations for each case up to the specified limit. These often involve breaking the problem into cases and using the addition principle.

28. What is the significance of the permutation matrix in linear algebra?

A permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. When multiplied with another matrix, it permutes the rows or columns of that matrix. This concept connects permutations to linear transformations and group theory.

29. How do permutations relate to group theory in abstract algebra?

In group theory, permutations form a fundamental example of a non-abelian group called the symmetric group. The study of permutation groups is crucial in understanding more complex algebraic structures and has applications in various areas of mathematics and physics.

30. How do you find the inverse of a permutation?

To find the inverse of a permutation, write the permutation in two-line notation with the original positions on top and the new positions below. Then, swap the rows and reorder the columns so that the top row is in ascending order. The bottom row now represents the inverse permutation.

31. What is the sign of a permutation?

The sign (or parity) of a permutation is either +1 (even permutation) or -1 (odd permutation). It's determined by whether the permutation can be achieved by an even or odd number of swaps of two elements. This concept is important in determinant calculations and group theory.

32. What is the role of permutations in generating functions?

Permutations play a role in certain types of generating functions, particularly in combinatorial problems. The exponential generating function for permutations is 1/(1-x), which encodes information about the number of permutations of each size in its coefficients.

33. How do permutations apply to the study of graph theory?

In graph theory, permutations are used to study graph isomorphisms and automorphisms. They help in determining when two graphs are essentially the same (isomorphic) or in finding the symmetries within a graph (automorphisms). This has applications in chemistry, computer science, and network analysis.

34. What is the connection between permutations and determinants in linear algebra?

The determinant of a matrix can be defined as a sum over all permutations of the matrix indices. Each term in this sum is a product of matrix elements, with a sign determined by the parity of the permutation. This definition links permutations directly to an important concept in linear algebra.

35. How do permutations relate to the concept of orbits in group actions?

In the study of group actions, permutations help describe how a group moves elements of a set. The orbit of an element is the set of all positions it can reach under the action of the group. Understanding these orbits often involves analyzing the permutations induced by the group action.

36. How do permutations relate to the concept of conjugacy classes in group theory?

In group theory, conjugacy classes of permutations are sets of permutations that have the same cycle structure. Understanding these classes is crucial for classifying elements in symmetric groups and other permutation groups. This concept links permutations to important structural properties of groups.

37. How do permutations relate to the Fundamental Counting Principle?

The Fundamental Counting Principle states that if one event can occur in m ways, and another independent event can occur in n ways, then the two events can occur together in m × n ways. This principle forms the basis for many permutation calculations.

38. How do you calculate the number of permutations when some objects are identical?

When some objects are identical, divide the total number of permutations by the factorial of the number of each identical object. For example, to find the permutations of the letters in "BOOKKEEPER": 10! / (2!3!2!2!) = 151,200.

39. What is the connection between permutations and the multiplication principle?

The multiplication principle is the foundation for permutation calculations. It states that if one event can happen in m ways, and another independent event can happen in n ways, then the two events can happen together in m × n ways. This principle is applied repeatedly in permutation problems.

40. How do permutations apply to cryptography?

Permutations are crucial in cryptography for creating and analyzing ciphers. They are used in encryption algorithms to rearrange data in a specific order, making it difficult for unauthorized parties to decipher the information without knowing the specific permutation used.

41. What is the difference between linear and circular permutations?

Linear permutations arrange objects in a line, where the first and last positions are distinct. Circular permutations arrange objects in a circle, where rotations of the same arrangement are considered identical. The number of circular permutations is always (n-1)! times fewer than linear permutations for n objects.

42. How do you handle permutations with repeated elements?

For permutations with repeated elements, divide the total number of permutations by the factorial of the number of repetitions for each repeated element. For example, the permutations of "PEPPER" would be 6! / (2!2!) = 180.

43. What is the lexicographic order of permutations?

Lexicographic order is a way to systematically list all permutations of a set. It's similar to dictionary order, where permutations are listed based on the order of their elements from left to right. This concept is important in algorithms for generating all permutations of a set.

44. What is the cycle notation in permutations?

Cycle notation is a compact way to represent permutations. It shows how elements move in cycles. For example, the permutation that moves 1 to 2, 2 to 3, and 3 to 1 is written as (123). This notation is particularly useful for composing permutations and finding their orders.

45. How do permutations apply to the solution of the Rubik's Cube?

Solving a Rubik's Cube involves a series of permutations of its faces and individual pieces. Understanding permutation groups helps in analyzing the possible states of the cube and developing efficient solving algorithms. The total number of possible cube states is a large permutation number.

46. How do permutations relate to the concept of symmetry in mathematics?

Permutations are fundamental in describing symmetries of mathematical objects. The symmetry group of an object consists of all permutations of its elements that preserve its structure. This concept links permutations to geometry, crystallography, and quantum mechanics.

47. What is the role of permutations in the study of permutation groups?

Permutation groups are groups whose elements are permutations of a set. They are fundamental in group theory and have applications in Galois theory, the study of polynomial equations, and symmetry analysis. Understanding permutations is crucial for grasping the structure and properties of these groups.

48. What is the significance of even and odd permutations in algebra?

The parity (evenness or oddness) of permutations is crucial in various algebraic contexts. It's used in the definition of the determinant, in the study of alternating groups (a special type of permutation group), and in understanding the solvability of polynomial equations of degree 5 or higher.

49. What is the significance of permutations in the representation theory of finite groups?

Permutation representations are fundamental in the representation theory of finite groups. They provide concrete ways to understand abstract group actions and are stepping stones to more complex representations. This connection highlights the importance of permutations in bridging concrete and abstract mathematics.

50. What is the role of permutations in the study of symmetric polynomials?

Symmetric polynomials are those that remain unchanged under any permutation of their variables. The theory of symmetric polynomials, which has applications in algebra and combinatorics, heavily relies on understanding how permutations act on polynomial expressions. This showcases the far-reaching influence of permutation concepts in advanced mathematics.