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Permutation

Last Updated : 13 Mar, 2024
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Permutation in mathematics is the arrangement of the object in a definite order. Permutation is similar to the combination and the basic difference between permutation and combination is that in permutation the order in which the object is taken is important. And the combination is the arrangement of the objects when the order of the objects is not important. Permutation is represented by the letter, P. For example permuation of set A = {1, 2, 3} when taken two object at a time is, {1, 2}, {1, 3}, {2, 3}, {3, 2}, {3, 1}, {2, 1}.

In this article, we will learn about, permutation, Permutation Formula, Examples, and others in detail.

What is a Permutation?

Permutation is defined as a mathematical calculation that tells us the way a particular set of data are arranged in a particular way. In simple word we can say that permutation is the number of ways objects can be ordered and arranged.

Permutation Definition

If n objects are available and we arrange all, then every arrangement possible is called a permutation.

In other words, arranging some objects in a specific order is called permutation. For example, we need to choose two numbers out of {1, 2, 3}, then there can be 6 ways in which we can do the same. (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), and (3, 1). As we can see here both (1, 2) and (2, 1) are two different permutations, unlike combinations where (1, 2) and (2, 1) are the same combination.

Representation of Permutation

There are various different representations of permutation, which mean the same thing. Some of the most common permutation representations or notations are as follows:

  • P(n, r)
  • nPr
  • nPr
  • Pn, k

Before learning more about Permutation we must know about the factorials. Let’s learn about the factorial in detail.

What is Factorial?

Factorial of a natural number n is denoted by the notation n! which represents the product of all natural numbers starting from 1 till n, including 1 and n. i.e. n! = n × (n-1) × (n-2) × (n-3) . . . × 1.

For example, If n = 3, then 3! = 3 × 2 × 1 = 6, and If n = 5 then 5! = 5 × 4 × 3 × 2 × 1 = 120.

Note: The factorial of 0 is defined as 1 by convention i.e.

0! = 1

Example of Permutation

Let’s consider another example of permutation if you have a set of three distinct elements: A, B, and C, them all possible permutations are:

ABC ACB BAC BCA CAB CBA

Permutation Examples in Real Life

Some real-life examples of Permutation include password generation, seating arrangement, shuffling cards, phone numbers, lock combination, etc. Some of the real-life examples representing permutations are illustrated in the following image added below,

Permutation

Properties of Permutations

Some of the common properties of permutations are listed as follows:

  • nPn = n (n-1) (n-2) . . . 1 = n!
  • nP0 = n! / n! = 1
  • nP1 = n
  • nPn – 1 = n !
  • nPr / nPr-1 = n – r + 1
  • nPr =n × n-1Pr-1 = n × (n-1) × n-2Pr-2 = n × (n-1) × (n-2) × n-3Pr-3 = and so on.
  • n-1Pr + r × n-1Pr-1 = nPr

Permutation Formula ( nPr )

The formula for Permutation is given as follows,

Permutation-Formula

where,

  • n is the Number of Total Objects
  • r is the Number of Objects Chosen at Once
  • 0 ≤ r ≤ n

Derivation of Permutation Formula

To derive the formula for permutation, we can use the first principle of counting and permutation in the following manner:

By the definition of permutation and principle of counting, we know

nPr = n × (n – 1)  . . .  (n – r + 1)

Multiplying and Dividing by (n – r)! on the LHS, we get

nPr = n × (n – 1) × (n – 2) × . . . × (n – r + 1) × (n – r)! / (n – r)!

nPr = n × (n – 1) × (n – 2) × . . . × (n – r + 1) × (n – r) × (n – r – 1) × . . . × 1 / (n – r)!

nPr = n! / (n – r)! where  0 ≤ r ≤ n 

Let’s consider some problems based on the derived formula to better understand its use.

Example 1: Find 6P3

As per the permutation formula,

6P3 = 6! / (3!)

= 6 × 5 × 4 = 120

Example 2: Find n if nP2 = 12

nPr = n! / (n – r)!

nP2 = n! / (n – 2)!

nP2 = n × (n – 1) × (n – 2)! / (n – 2)!

nP2 = n × (n – 1)

nP2 = n2 – n

∴ n2 – n = 12

Solving the equation,

n2 – n – 12 = 0

⇒ n (n – 4) + 3 (n – 4) = 0

⇒ (n + 3) (n – 4) = 0

∴ n = -3 or n = 4

∵ n ≥ 0

Thus, n = 4

Types of Permutation

In the study of permutation, there are some cases such as:

  • Permutation with Repetition
  • Permutation without Repetition
  • Permutation of Multi-Sets

Let’s explain these cases in detail with solved example as follows:

Permutation With Repetition

This is the simplest of the lot. In such problems, the objects can be repeated. Let’s understand these problems with some examples.

Example: How many 3-digit numbers greater than 500 can be formed using 3, 4, 5, and 7?

Since a three-digit number, greater than 500 will have either 5 or 7 at its hundredth place, we have 2 choices for this place.

There is no restriction on repetition of the digits, hence for the remaining 2 digits we have 4 choices each

So the total permutations are,

2 × 4 × 4 = 32

Permutation Without Repetition

In this class of problems, the repetition of objects is not allowed. Let’s understand these problems with some examples.

Example: How many 3-digit numbers divisible by 3 can be formed using digits 2, 4, 6, and 8 without repetition?

For a number to be divisible by 3, the sum of it digits must be divisible by 3

From the given set, various arrangements like 444 can be formed but since repetition isn’t allowed we won’t be considering them.

We are left with just 2 cases i.e. 2, 4, 6 and 4, 6, 8

Number of arrangements are 3! in each case

Hence the total number of permutations are: 3! + 3! = 12

Permutation of Multi-Sets

Permutation when the objects are not distinct

This can be thought of as the distribution of n objects into r boxes where the repetition of objects is allowed and any box can hold any number of objects.

1st  box can hold n objects

2nd box can hold n objects

3rd box can hold n objects
.                                          . 
.                                          . 
.                                          . 

rth box can hold n objects

Hence total number of arrangements are,

n × n × n . . . (r times) = nr

Examples: A police officer visits the crime scene 3 times a week for investigation. Find the number of ways to schedule his visit if there is no restriction on the number of visits per day?

Number of ways to schedule first visit is 7 (any of the 7 days)

Number of ways to schedule second visit is 7 (any of the 7 days)

Number of ways to schedule third visit is 7 (any of the 7 days)

Hence, the number of ways to schedule first and second and third visit is

7 × 7 × 7 = 73 = 343

Permutation and Combination

Permutation and Combination are the two most fundamental concepts in combinatorics, as these lay the foundation of this branch of mathematics. Combinatorics deals with all the mathematics which we need to solve problems based on selection, arrangement, and operation within a finite or discrete system.

Relation Between nPr and nCr

We can understand nCr through the following analogy. Consider that we have n distinct boxes and r identical balls. (n > r)

The task is to place all the r balls into boxes such that no box contains more than 1 ball.

As all r objects are the same, the r! ways of arranging them can be considered as a single way.

To group all r! ways of arranging, we divide nPr by r!

nCr = nPr/r! = n!/{(n – r)! × r!}

Hence the relation between nPr and nCr is,

nCr = nPr/r!

Permutation Vs Combination

The key differences between permutation and combination, some of those differences are listed as follows:

Aspect

Permutations

Combinations

Definition Arrangements of elements in a specific order. Selections of elements without considering the order.
Formula nPr = n!​/(n−r)! nPr = n!​/[(n−r)! × r!]
Notation nPr  OR P(n, r) nCr  OR C(n, r)
Order Matters Yes, order matters. No, order doesn’t matter.
Example Arranging books on a shelf. Selecting members for a committee.
Sample Problems How many ways to arrange 3 books out of 5? How many ways to choose 2 fruits from a basket of 7?
Application Permutations are used when order matters, 
such as arranging items in a sequence or forming a code.
Combinations are used when order doesn’t matter, 
like selecting a group of people or choosing items without caring about their order.

Learn more about Permutation Vs Combination

Fundamental Counting Principle

Fundamental principle of the counting states that, “To perform a operation if we have m ways and to perform second operation if we have n different ways then to perform both the operation together there are m × n different ways.”

We can also extend the principle of counting for more operation such that, if for 1 operation we have m, 2 operation we have n, for third operation we have o, different ways respectively than, for performing all the operation together we have m × n × o ways.

Read More,

Problems on Permutation

Problem 1. How many 4-letter words, with or without meaning can be formed out of the letters of the word, ‘SATURDAY’ if repetition of letters is not allowed?

Solution:

Word SATURDAY has 8 letters i.e. S, A, T, U, R, D, A, and Y

To form 4-letter words, we first have to select 4 letters from these 8 letters

The ways of selecting 4 letters from 8 letters disregarding the order is 8C4 .

After selection, there are 4! arrangements.

Hence, total number of words formed are: 8C4 × 4!

Note: Selecting r objects out of n objects and then arranging them is same as r-permutation of n objects.

Problem 2. Find the number of ways of selecting 6 balls from 4 red, 6 blue, and 5 white given that the selection must have 2 balls of each color?

Solution:

We need to select 2 balls each of color red, blue and white as per the given condition.

Number of ways of selecting 2 red balls is 4C2

Number of ways of selecting 2 blue balls is 6C2

Number of ways of selecting 2 white balls is 5C2

Hence, the total ways of selection are 4C2 × 6C2 × 5C2  = 900

Problem 3. A class has just 3 seats vacant. Three people P, A, and R arrive at the same time. In how many ways can P, A, and R be arranged on those 3 vacant seats?

Solution:

For the very first seat, we have 3 choices i.e. P, A and R.

Let us randomly select A for the first seat.

For the second seat, we have 2 choices i.e. P and R

Let us randomly select R for the second seat.

For the third seat, we have 1 choice i.e. P

To summarize, we did the following:

Placed a person on seat 1 and placed a person on seat 2 and placed a person on seat 3.

Usage of and comes from the fact that occupation of all 3 seats was mandatory.

In mathematics, and is related with multiplication, hence we can say that total choices = 3 × 2 × 1 = 3!

If we change the seating order to P on the first seat, A on the second seat, and R on the third, does that change the total number of choices?

No, it does not. This is because equal importance is given to all three P, A, and R.

Problem 4. Find the number of ways of arranging 5 people if 2 of them always sit together?

Solution:

Let us consider the 2 people as a unit and the remaining 3 person as 3 separate units, So we have total 4 units.

Number of ways of arranging these 4 units is 4!

(just the way we proved in previous problem)

Number of ways of arranging the 2 person amongst themselves is 2!

In conclusion, the number of ways of arranging the 4 units and 2 person amongst themselves is 4! × 2!

Problem 5. Find all the three-letter words beginning and ending with a vowel. Given that repetition of alphabets is not allowed.

Solution:

Total vowels in english = 7 ( a, e, i, o, u, y, w)

Total consonants in english = 26 – 7 = 19

  • Choices for the first letter are 7
  • Choices for the third letter are 6 (since 1 vowel was placed as first letter)
  • Choices for the middle letter are 19 + (7 – 2) = 24 (19 consonants + the vowels which were not placed)

Hence, total permutations are 7 × 6 × 24 = 1008

Do observe that here we first satisfied the vowel condition for first and third letter and then placed the middle letter.

Problem 6. An ice-cream shop has 10 flavors of ice cream. Find the number of ways of preparing an ice cream cone with any 3 different flavors?

Solution:

Let us consider n = 10 (total number of flavors) and r = 3 (number of different flavors needed)

For first flavor we have 10 choices

For second flavor we have 10 – 1 choices

For third flavor we have 10 – 2 choices and this is same as (n – r + 1)

The numbers of arrangement would be: 10 × (10 – 1) × (10 – 3 + 1) = 720

From this we can generalize that, the number of ways of arranging r objects out of n different objects is:

n × (n – 1)  . . . (n – r + 1) = nPr

Problem 7. 10 Olympians are running a race. Find the different arrangements of 1st, 2nd, and 3rd place possible?

Solution:

We have to find different arrangements of 10 taken 3 at time.

Here,

  • n = 10
  • r = 3

Different arrangement of for 1, 2, and 3 places are

10P3 = 10! / (7!)

= 10 × 9 × 8 = 720

Problem 8. How many even numbers lying between 1000 and 2000 can be formed using the digits 1, 2, 4, 5, and 9?

Solution:

Since the number is supposed to be even, the digits at units place must either be 2 or 4 leaving us with 2 choices for the digit at units place.

The number is supposed to lie between 1000 and 2000, So the digits at thousand’s place must be 1, we thus have 

1 choice for the digit at thousands place.

Rest of the 2 digits can be any one from 1, 2, 4, 5 and 9 i.e. 5 choices each

So the total permutations are: 2 × 5 × 5 × 1 = 50

Problem 9. How many 4 digit numbers divisible by 5 can be formed using 0, 3, 5, 7, and 9 if repetition of digits is not allowed?

Solution:

For the number to be divisible by 5, the digit at units place must either be 0 or 5, hence we have 2 possibilities.

Case 1. Digit at units place is 0

There are 4 choices for 103 place (all numbers except 0)

There are 3 choices for the 102 place (1 got used up at 103 place)

There are 2 choices for the 101 place (1 got used up at 103 place and 1 at 102 place)

Hence the possible arrangements with 0 at units place are

4 × 3 × 2 = 24

Case 2. Digit at units place is 5

There are 3 choices for 103 place (all except 0 and 5)

There are 2 choices for 102 place (1 got used up at 103 place)

There is 1 choice for 101 place (1 got used up at 103 place and 1 at 102 place)

Hence the possible arrangements with 5 at units place are 3 × 2 × 1 = 6

Total Arrangements = Number of arrangements in case 1 + Number of arrangements in case 2

Total Arrangements = 24 + 6 = 30

Practice Questions on Permutation

Q1. In how many ways can 6 prisoners be placed in 4 cells if any number of prisoners can fit in a cell?

Q2. Find how many 4 digit numbers divisible by 8 can be formed using 0, 1, 2, 3, 5, 7, and 9 if repetition of digits is not allowed?

Q3. Find the number of ways of selecting 8 balls from 10 red, 16 blue, and 15 white given that the selection must have 1 balls of each color?

Q4. Find how many even numbers lying between 4000 and 8000 can be formed using the digits 1, 2, 3, 4, 5, and 6 when repetition is allowed?

FAQs on Permutation

1. What Does a Permutation Mean?

Permutation means arranging some certain items in specific order.

2. What is Permutation class 11?

In mathematics, permutation is the mathematical calulation for finding the number of ways to arrenge some object in any specific order.

3. What does “n!” mean?

“n!” (read as “n factorial”) represents the product of all positive integers from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

4. What is Formula for Permutation?

The permutation formula for n objects taken r at a time is,

nPr = n!/(n – r)!

5. How do I Calculate the Number of Permutations of a Set of Objects?

The number of permutations of a set of n distinct objects is given by n!.

6. How is a Permutation Different from a Combination?

A permutation considers the arrangement of elements in a specific order, while a combination only considers the selection of elements without considering the order.

7. What is Circular Permutation?

A circular permutation is a type of arrangement where the order of elements matters, but the arrangement is considered circular. That is, the first and last elements are treated as adjacent.

8. What Are the 4 Types of Permutations?

The 4 types of the perutations are,

  • Permutations with Repetition
  • Permutation without Repetition
  • Permutations with Multi-Sets
  • Circular Permutations

9. What’s the Difference Between a Permutation and Combination?

The basic difference between permutation and combination is, that in permutation order of object is important and in the combination the order of the object is not important.

10. What is the Permutation for Multisets?

Permutation for Multisets formula is,

P(n, r)(Multiset) = n!/(P1!P2!…Pn!)



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