# Permutations Formula: Definition, Types, Usage, Principle, Concepts, Solved Examples

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Posted by Osman Gezer, 2/19/24

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Permutations are an essential concept in mathematics and combinatorics, allowing us to calculate the number of different arrangements or orders in which a set of objects can be arranged. The permutations formula provides a systematic and efficient way to determine these arrangements. In this article, we will explore the definition, types, usage, principle, and concepts of permutations formula, along with solved examples to illustrate its application.

## An Introduction to Permutations Formula

In mathematics, permutations refer to the arrangement of objects in a specific order or sequence. Each element or member of the set is arranged in a distinct and well-defined position. The permutations formula allows us to calculate the number of possible arrangements of a set of objects, taking into account the order in which they are arranged.

Permutations play a crucial role in various fields of mathematics, statistics, and combinatorics. They are used to solve problems related to arranging, selecting, or ordering objects or elements from a given set. Permutations are widely applicable in real-life scenarios, such as transportation schedules, license plate codes, and event seating arrangements.

## What is Permutation?

A permutation is the arrangement of a set of objects in a specific order or sequence. It involves selecting and arranging items from a set in a particular sequence. In a permutation, changing the order of selected items results in a different permutation. For example, arranging the letters of the word “ABC” in different orders like ABC, BCA, CAB, etc., are distinct permutations.

## What is the Permutations Formula?

The permutations formula is a mathematical representation used to calculate the number of ways to arrange objects in a specific order. It is denoted as P(n, r), where “n” represents the total number of objects in the set, and “r” represents the number of objects to be selected and arranged.

The formula for calculating permutations is:

P(n, r) = n! / (n – r)!

Here, “!” denotes the factorial of a number, which is the product of all positive integers from 1 to that number.

## Permutation Formula

The permutation formula is used to find the number of ways of selecting and arranging “r” different things from “n” different things. It calculates the number of permutations without repetition.

The formula for permutation is:

nPr (or) P(n, r) = (n!) / (n – r)!

In this formula:

• “n” represents the total number of things.
• “r” represents the number of things to be selected and then arranged.

The permutation formula is derived from the concept of factorial. The factorial of any number is the product of consecutive numbers starting from 1 and ending at that number.

Formula 1: Factorial of a natural number n. n! = 1 × 2 × 3 × 4 × …….× n

Formula 2: Permutation Formula or NPR formula for “r” things taken from “n” things. nPr = (n!) / (n – r)!

Formula 3: The relationship between permutations and combinations for “r” things taken from “n” things. nPr = r! × nCr

## How to Calculate Permutations?

Calculating permutations using the permutation formula involves the following steps:

Step 1: Identify the total number of things (n) and the number of things to be selected and arranged (r).

Step 2: Substitute these values into the permutation formula: nPr = (n!) / (n – r)!

Step 3: Calculate the factorial of “n” by multiplying all the consecutive numbers from 1 to “n”.

Step 4: Calculate the factorial of the difference between “n” and “r” by multiplying all the consecutive numbers from 1 to (n – r).

Step 5: Divide the factorial of “n” by the factorial of (n – r) to get the total number of permutations.

Step 6: Simplify the expression if possible.

Step 7: The result obtained is the total number of permutations.

## How to Use the Permutations Formula?

The permutations formula can be used in various scenarios where the arrangement or order of objects is important. Here are some common applications of the permutations formula:

1. Arranging a set of objects in a specific order: The permutations formula helps in calculating the number of ways to arrange a set of objects in a specific order. For example, finding the number of different words that can be formed using a given set of letters without repetition.
2. Seating arrangements: The permutations formula is useful in determining the number of possible seating arrangements for a group of people. It helps in calculating the number of ways individuals can be arranged in a row or around a table.
3. Event scheduling: The permutations formula is used in planning the schedules for events, such as train departures, bus routes, or flight timings. It helps in determining the number of possible arrangements for the convenience of the public.
4. Password generation: The permutations formula is applied in generating unique passwords by considering the different combinations of characters, digits, and symbols.
5. License plate codes: The permutations formula enables the creation of unique license plate codes by arranging alphabets and digits in different orders.

## When to Use the Permutations Formula?

The permutations formula is used when the order or arrangement of objects is important. Here are some scenarios where the permutations formula is applicable:

1. Arranging a set of items in a specific order without repetition: When you need to calculate the number of ways to arrange a set of items without repetition, such as arranging letters to form words or arranging numbers to create sequences.
2. Determining the number of possible outcomes in a sequence: When you want to calculate the number of possible outcomes in a specific sequence, such as determining the number of different ways to arrange the positions in a race.
3. Planning seating arrangements: When you need to plan seating arrangements for events, gatherings, or classrooms, the permutations formula helps in calculating the number of possible arrangements.
4. Creating unique codes or passwords: When generating unique codes or passwords, the permutations formula helps in calculating the number of possible combinations of characters, digits, and symbols.

Overall, the permutations formula is used in situations where the order or arrangement of objects is important, and it helps in determining the number of possible outcomes.

## Fundamental Counting Principle

The Fundamental Counting Principle is a concept closely related to permutations. It is used to calculate the total number of outcomes when making multiple sequential choices or arranging items. The principle states that if there are “n” ways to make the first choice and “m” ways to make the second choice, then there are “n x m” ways to make both choices sequentially.

The Fundamental Counting Principle is particularly useful when each choice affects the subsequent ones. It helps in calculating the total number of permutations when arranging elements in specific orders, where the choices made for each position are not independent.

The principle can be summarized as follows:

“If there are ‘n’ ways to make the first choice and ‘m’ ways to make the second choice, then there are ‘n x m’ ways to make both choices sequentially.”

In general, for a permutation with “r” positions, if there are “n1” choices for the first position, “n2” choices for the second position, “n3” choices for the third position, and so on, the total number of permutations is:

Total Permutations = n1 x n2 x n3 x … x nr

The Fundamental Counting Principle is a fundamental concept in combinatorics and is widely used in various real-life scenarios where sequential choices or arrangements are involved.

## Derivation of Permutation Formula

The permutation formula is derived from the concept of factorial and the Fundamental Counting Principle. Let’s understand the derivation of the permutation formula step by step.

Step 1: Consider a set of “n” distinct objects.

Step 2: To calculate the number of permutations when selecting “r” objects, we start by considering the number of choices for the first position. Since there are “n” objects in the set, there are “n” choices for the first position.

Step 3: After making the first choice, we move on to the second position. Here, the number of choices decreases by 1, as we have already chosen one object for the first position. Therefore, there are (n-1) choices for the second position.

Step 4: Continuing this process, for the third position, there are (n-2) choices, for the fourth position, there are (n-3) choices, and so on, until the rth position.

Step 5: According to the Fundamental Counting Principle, the total number of permutations is obtained by multiplying the number of choices for each position.

Therefore, the number of permutations can be calculated as:

P(n, r) = n × (n-1) × (n-2) × … × (n-(r-1))

This can be further simplified as:

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

Hence, the permutation formula is derived.

## Types of Permutation Formula

There are different types of permutation formulas, depending on the specific characteristics of the arrangements. Let’s explore each type in detail.

### Permutations Formula without Repetition

This type of permutation involves arranging a set of “n” distinct objects in a specific order, where each object appears only once in the arrangement, and repetitions are not allowed. The number of permutations without repetition is calculated using the permutation formula:

nPr (or) P(n, r) = (n!) / (n – r)!

For example, if we have a set of 5 letters (A, B, C, D, E) and want to find the number of 3-letter words that can be formed without repetition, we can use the permutation formula:

n = 5 (total number of letters) r = 3 (number of letters in each word)

nPr = 5P3 = (5!) / (5 – 3)! = 5! / 2! = (5 × 4 × 3 × 2 × 1) / (2 × 1) = 60

Therefore, there are 60 ways to form 3-letter words without repetition using the given set of letters.

### Permutations Formula with Repetition

In this type of permutation, repetitions are allowed. It means that you can arrange a set of objects where some or all of the objects can appear more than once in the arrangement. The number of permutations with repetition can be calculated using exponent notation:

n × n × n × … (repeated “r” times) = nr

For example, if we have a set of 5 letters (A, B, C, D, E) and want to find the number of 3-letter words that can be formed with repetition, we can use the permutation formula:

n = 5 (total number of letters) r = 3 (number of letters in each word)

nPr = 5P3 = 53 = 125

Therefore, there are 125 ways to form 3-letter words with repetition using the given set of letters.

### Permutations Formula Taken All at a Time

The number of ways of arranging “n” different things among themselves is known as permutations taken all at a time. It refers to arranging all “n” objects out of “n” objects and is given by:

nPn = n! / (n-n)! = n!/0! = n!

For example, if we have 5 different books in a bookshelf, we can calculate the number of ways to arrange them using the permutations formula:

n = 5 (total number of books)

nPn = 5P5 = 5! = 5 × 4 × 3 × 2 × 1 = 120

Therefore, there are 120 ways to arrange the 5 different books on the bookshelf.

### Permutations Formula with Same Sets of Data

In some cases, not all objects in a set are different, and some may be the same. In such scenarios, the number of possible arrangements can be calculated using the permutations formula with the same sets of data. If there are “r” same things, “s” same things, and “p” same things out of “n” total things, the number of possible arrangements is given by:

n! / (r! × s! × p!)

For example, let’s find the number of arrangements of the letters in the word “PETER”:

Number of letters in the given word = 5 Among them, ‘E’ is repeated 2 times

So the possible number of arrangements = 5! / 2! = (120)/2 = 60

Therefore, there are 60 different arrangements of the letters in the word “PETER”.

### Circular Permutations Formula

All the previous formulas refer to arrangements in a line. But when it comes to arranging “n” different objects in a circle, the number of ways is (n – 1)!. This is known as circular permutations.

For example, if we want to find the number of ways to arrange 10 people around a round table, we can use the circular permutations formula:

The required number of ways = (10-1)! = 9!

Therefore, there are 9! or 362,880 ways to arrange 10 people around a round table.

These are the different types of permutation formulas that can be applied in various scenarios depending on the arrangement requirements and characteristics of the objects.

## Solved Examples on Permutations Formula

Let’s solve some examples to illustrate the application of the permutations formula.

Example 1: Find the number of ways in which a three-digit code can be formed using the numbers 4, 5, 7, 8, 9.

Solution: The given digits are 4, 5, 7, 8, 9.

Total number of digits (n) = 5 Number of digits in the code (r) = 3

Applying the permutation formula, we have: The number of ways of forming a 3-digit code = 5P3 = (5!) / (5 – 3)! = 5! / 2! = (5 × 4 × 3 × 2 × 1)/(2 × 1) = 5 × 4 × 3 = 60

Answer: Hence, there are 60 ways of forming a three-digit code using the given digits.

Example 2: How many different words with or without meaning can be formed using any 4 letters from a word containing 10 different letters?

Solution: Total number of letters (n) = 10 Number of letters to form the new word (r) = 4

Using the permutations formula, we have: Total number of 4-letter words that can be formed = 10P4 = 10P4 = (10!) / (10-4)! = 10!/6! = (10 × 9 × 8 × 7 × 6!)/6! = 10 × 9 × 8 × 7 = 5040

Answer: Hence, a total of 5040 four-letter words can be formed.

These solved examples demonstrate the practical application of the permutations formula in solving real-life problems related to arranging objects, forming codes, or creating new words.