# Permutation Formula

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## What Is a Permutation?

A permutation is a mathematical calculation of the number of ways a particular set can be arranged, where the order of the arrangement matters.

## Formula and Calculation of Permutation

The formula for a permutation is:

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

where

n = total items in the set; r = items taken for the permutation; “!” denotes factorial

The generalized expression of the formula is, “How many ways can you arrange ‘r’ from a set of ‘n’ if the order matters?” A permutation can be calculated by hand as well, where all the possible permutations are written out. In a combination, which is sometimes confused with a permutation, there can be any order of the items.

## What Permutation Can Tell You

A simple approach to visualize a permutation is the number of ways a sequence of a three-digit keypad can be arranged. Using the digits 0 through 9, and using a specific digit only once on the keypad, the number of permutations is P(10,3) = 10! / (10-3)! = 10! / 7! = 10 x 9 x 8 = 720. In this example, order matters, which is why a permutation produces the number of digit entryways, not a combination.

In finance and business, here are two examples. First, suppose a portfolio manager has screened out 100 companies for a new fund that will consist of 25 stocks. These 25 holdings will not be equal-weighted, which means that ordering will take place. The number of ways to order the fund will be: P(100,25) = 100! / (100-25)! = 100! / 75! = 3.76E + 48. That leaves a lot of work for the portfolio manager to construct his fund!

An easier example would be, say a company wants to build out its warehouse network across the country. The company will commit to three locations out of five possible sites. Order matters because they will be built sequentially. The number of permutations is: P(5,3) = 5! / (5-3)! = 5! / 2! = 60.

## Permutation

A permutation is an arrangement of objects in a definite order. The members or elements of sets are arranged here in a sequence or linear order. For example, the permutation of set A={1,6} is 2, such as {1,6}, {6,1}. As you can see, there are no other ways to arrange the elements of set A.

In permutation, the elements should be arranged in a particular order whereas in combination the order of elements does not matter. Also, read: Permutation And Combination

When we look at the schedules of trains, buses and the flights we really wonder how they are scheduled according to the public’s convenience. Of course, the permutation is very much helpful to prepare the schedules on departure and arrival of these. Also, when we come across licence plates of vehicles which consists of few alphabets and digits. We can easily prepare these codes using permutations.

## Definition of Permutation

Basically Permutation is an arrangement of objects in a particular way or order. While dealing with permutation one should concern about the selection as well as arrangement. In Short, ordering is very much essential in permutations. In other words, the permutation is considered as an ordered combination.

## Representation of Permutation

We can represent permutation in many ways, such as:

## Formula

The formula for permutation of n objects for r selection of objects is given by:
P(n,r) = n!/(n-r)!
For example, the number of ways 3rd and 4th position can be awarded to 10 members is given by:
P(10, 2) = 10!/(10-2)! = 10!/8! = (10.9.8!)/8! = 10 x 9 = 90

## Types of Permutation

Permutation can be classified in 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 multi sets)

Let us understand all the cases of permutation in details.

### 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:

Here, “nPr” represents the “n” objects to be selected from “r” objects without repetition, in which the order matters.

Example: How many 3 letter words with or without meaning can be formed out of the letters of the word SWING when repetition of letters is not allowed?

Solution: Here n = 5, as the word SWING has 5 letters. Since we have to frame 3 letter words with or without meaning and without repetition, therefore total permutations possible are:

### Permutation when repetition is allowed

We can easily calculate the permutation with repetition. The permutation with repetition of objects can be written using the exponent form.

When the number of object is “n,” and we have “r” to be the selection of object, then;

Choosing an object can be in n different ways (each time).

Thus, the permutation of objects when repetition is allowed will be equal to,

n × n × n × ……(r times) = nr

This is the permutation formula to compute the number of permutations feasible for the choice of “r” items from the “n” objects when repetition is allowed.

Example: How many 3 letter words with or without meaning can be formed out of the letters of the word SMOKE when repetition of words is allowed?

Solution:

The number of objects, in this case, is 5, as the word SMOKE has 5 alphabets.

and r = 3, as 3-letter word has to be chosen.

Thus, the permutation will be:

Permutation (when repetition is allowed) = 53 = 125

### Permutation of multi-sets

Permutation of n different objects when

### Difference Between Permutation and Combination

The major difference between the permutation and combination are given below:

## Fundamental Counting Principle

According to this principle, “If one operation can be performed in ‘m’ ways and there are n ways of performing a second operation, then the number of ways of performing the two operations together is m x n “.

This principle can be extended to the case in which the different operation be performed in m, n, p, . . . . . . ways.

In this case the number of ways of performing all the operations one after the other is m x n x p x . . . . . . . . and so on

### Solved Examples

 Example 1: In how many ways 6 children can be arranged in a line, such that (i) Two particular children of them are always together (ii) Two particular children of them are never together Solution: (i) The given condition states that 2 students need to be together, hence we can consider them 1. Thus, the remaining 7 gives the arrangement in 5! ways, i.e. 120. Also, the two children in a line can be arranged in 2! Ways. Hence, the total number of arrangements will be, 5! × 2! = 120 × 2 = 240 ways (ii) The total number of arrangements of 6 children will be 6!, i.e. 720 ways. Out of the total arrangement, we know that two particular children when together can be arranged in 240 ways. Therefore, total arrangement of children in which two particular children are never together will be 720 – 240 ways, i.e. 480 ways.

Example 2:Consider a set having 5 elements a,b,c,d,e. In how many ways 3 elements can be selected (without repetition) out of the total number of elements.

Solution: Given X = {a,b,c,d,e}

3 are to be selected.

Therefore,

5C3 =10

Example 3: It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?

Solution: We are given that there are 5 men and 4 women.

i.e. there are 9 positions.

The even positions are: 2nd, 4th, 6th and the 8th places

These four places can be occupied by 4 women in P(4, 4) ways = 4!

= 4 . 3. 2. 1

= 24 ways

The remaining 5 positions can be occupied by 5 men in P(5, 5) = 5!

= 5.4.3.2.1

= 120 ways

Therefore, by the Fundamental Counting Principle,

Total number of ways of seating arrangements = 24 x 120

= 2880

### Practice Problems

Practice below listed problems:

1. How many numbers lying between 100 and 1000 can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed.
2. Seven athletes are participating in a race. In how many ways can the first three prizes be won.

To solve more problems or to take a test, download BYJU’S – The Learning App.

## Frequently Asked Questions – FAQs

### What is permutation?

Permutation is a way of changing or arranging the elements or objects in a linear order.

### What is the formula for permutation?

The formula for permutation for n objects taken r at a time is given by:
P(n,r) = n!/(n-r)!

### What are the types of permutation?

The permutation of an arrangement of objects or elements in order, depends on three conditions:
When repetition of elements is not allowed
When repetition of elements is allowed
When the elements of a set are not distinct

### What is the formula for permutation when repetition is allowed?

Let n be the number of objects and r be the selection of objects, then if repetition is allowed, the permutation of objects will be n × n × n × ……(r times) = n^r

### What is the permutation for multisets?

The permutation formula for multisets where all the elements are not distinct is given by: n!/(P1!P2!…Pn!)

## Permutations vs. Combinations

Both permutation and combinations involve a group of numbers. However, with permutations the order of the numbers matters. With combinations, the ordering does not matter.  For example, with permutation, the order matters, such as the case with a locker combination.

Locker combos are, thus, not combinations. They are permutations. A locker combo must be entered exactly as scripted, such as 6-5-3, or it will not work. If it were a true combination then the numbers could be entered in any order and work.

There are various types of permutations as well. You can find the number of ways of writing a group of numbers. But you can also find permutations with repetition. That is, the total number of permutations when the numbers can be used more than once or not at all.

## Permutation And Combination

Permutation and combination are the ways to represent a group of objects by selecting them in a set and forming subsets. It defines the various ways to arrange a certain group of data. When we select the data or objects from a certain group, it is said to be permutations, whereas the order in which they are represented is called combination. Both concepts are very important in Mathematics.

Permutation and combination are explained here elaborately, along with the difference between them. We will discuss both the topics here with their formulas, real-life examples and solved questions.  Students can also work on Permutation And Combination Worksheet to enhance their knowledge in this area along with getting tricks to solve more questions.

## What is Permutation?

In mathematics, permutation relates to the act of arranging all the members of a set into some sequence or order. In other words, if the set is already ordered, then the rearranging of its elements is called the process of permuting. Permutations occur, in more or less prominent ways, in almost every area of mathematics. They often arise when different orderings on certain finite sets are considered.

## What is a Combination?

The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection or k-combination with repetition are often used. Permutation and Combination Class 11 is one of the important topics which helps in scoring well in Board Exams.

## Permutation and Combination Formulas

There are many formulas involved in permutation and combination concepts. The two key formulas are:

### Permutation Formula

A permutation is the choice of r things from a set of n things without replacement and where the order matters.

nPr = (n!) / (n-r)!

### Combination Formula

A combination is the choice of r things from a set of n things without replacement and where order does not matter.

## Uses of Permutation and Combination

A permutation is used for the list of data (where the order of the data matters) and the combination is used for a group of data (where the order of data doesn’t matter).

## Solved Examples of Permutation and Combinations

Example 1:

Find the number of permutations and combinations if n = 12 and r = 2.

Solution:

Given,

n = 12
r = 2

Using the formula given above:

Permutation:

nPr = (n!) / (n-r)! =(12!) / (12-2)! = 12! / 10! = (12 x 11 x 10! )/ 10! = 132

Combination:

Example 2:

In a dictionary, if all permutations of the letters of the word AGAIN are arranged in an order. What is the 49th word?

Solution:

This accounts up to the 48th word. The 49th word is “NAAGI”.

Example 3:

In how many ways a committee consisting of 5 men and 3 women, can be chosen from 9 men and 12 women?

Solution:

Choose 5 men out of 9 men = 9C5 ways = 126 ways

Choose 3 women out of 12 women = 12C3 ways = 220 ways

The committee can be chosen in 27720 ways.

## Permutation and Combination – Practice Questions

Question 1: In how many ways can the letters be arranged so that all the vowels come together? Word is “IMPOSSIBLE.”

Question 2: In how many ways of 4 girls and 7 boys, can be chosen out of 10 girls and 12 boys to make the team?

Question 3: How many words can be formed by 3 vowels and 6 consonants taken from 5 vowels and 10 consonants?

## Frequently Asked Questions on Permutations and Combinations

### What do you mean by permutations and combinations?

A permutation is an act of arranging the objects or numbers in order.
Combinations are the way of selecting the objects or numbers from a group of objects or collection, in such a way that the order of the objects does not matter.

### Give examples of permutations and combinations.

An example of permutations is the number of 2 letter words that can be formed by using the letters in a word say, GREAT; 5P_2 = 5!/(5-2)!
An example of combinations is in how many combinations we can write the words using the vowels of the word GREAT; 5C_2 =5!/[2! (5-2)!]

### What is the formula for permutations and combinations?

The formula for permutations is: nPr = n!/(n-r)!
The formula for combinations is: nCr = n!/[r! (n-r)!]

### What are the real-life examples of permutations and combinations?

Arranging people, digits, numbers, alphabets, letters, and colours are examples of permutations.
Selection of menu, food, clothes, subjects, the team are examples of combinations.

### Write the relation between permutations and combinations.

The formula for permutations and combinations are related as:
nCr = nPr/r!

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