Permutation is a concept in mathematics that deals with the arrangement of objects in a specific order. It is a fundamental concept used in various fields, including mathematics, computer science, and statistics. Permutations can be thought of as different ways of arranging items or elements in a set. Each arrangement is unique and distinct, and the order of the elements matters.

## What is Permutation?

In simple terms, **permutation **is the rearrangement of objects in a specific order. It can be expressed as the number of ways to arrange *n* distinct objects, taken *r* at a time. The symbol used to denote the number of permutations is nPr, where *n* represents the total number of objects and *r* represents the number of objects taken at a time.

Permutations are commonly used in real-life situations such as scheduling, allocation, and coding. For example, permutations are used to schedule buses, trains, or flights, allocate zip codes and phone numbers, and generate unique codes or passwords.

## Permutation Formula

The formula to calculate permutations is given by:

```
nPr = n! / (n - r)!
```

Where:

`n`

is the total number of objects`r`

is the number of objects taken at a time`n!`

represents the factorial of`n`

, which is the product of all positive integers less than or equal to`n`

The factorial of a number is calculated by multiplying all positive integers less than or equal to that number. For example, `5!`

(read as “5 factorial”) is calculated as `5 × 4 × 3 × 2 × 1 = 120`

.

## Permutation Notation

Permutations are denoted using the symbol `nPr`

. The value of `n`

represents the total number of objects, and the value of `r`

represents the number of objects taken at a time. The notation `nPr`

is read as “n permute r.”

## How to Calculate Permutations?

To calculate permutations, follow these steps:

**Step 1**: Determine the total number of objects (`n`

) and the number of objects taken at a time (`r`

).

**Step 2**: Calculate the factorial of `n`

and `n-r`

.

**Step 3**: Divide the factorial of `n`

by the factorial of `n-r`

.

**Step 4**: The result is the number of permutations (`nPr`

).

Let’s take an example to understand the calculation of permutations.

**Example:** Suppose we have 5 different objects and we want to arrange them in groups of 3. To calculate the number of permutations, we can use the formula:

```
nPr = n! / (n - r)!
```

In this case, `n = 5`

and `r = 3`

. Plugging these values into the formula:

```
5P3 = 5! / (5 - 3)!
= 5! / 2!
= (5 × 4 × 3 × 2 × 1) / (2 × 1)
= 60
```

Therefore, there are 60 different permutations of 5 objects taken 3 at a time.

## Properties of Permutation

Permutations have several properties that are useful in solving permutation problems. Some of the important properties of permutations are:

- The number of permutations of
`n`

objects taken all at a time is`n!`

, which is the factorial of`n`

. - The number of permutations of
`n`

objects taken`r`

at a time, where`r <= n`

, is given by the formula`nPr = n! / (n - r)!`

. - If
`r = n`

, then`nPr = n! / (n - n)! = n! / 0! = n! / 1 = n!`

. - If
`r = 0`

, then`nPr = n! / (n - 0)! = n! / n! = 1`

.

These properties help in simplifying permutation calculations and understanding the concepts better.

## Alternate Analysis of nPr

An alternate way to analyze `nPr`

is to think of it as a step-by-step process. Let’s consider an example to illustrate this:

**Example:** Suppose we have 5 objects and we want to arrange 3 of them. We can think of the process as follows:

- For the first object, we have 5 choices.
- For the second object, we have 4 choices remaining (since we have already chosen one object).
- For the third object, we have 3 choices remaining (since we have already chosen two objects).

To find the total number of permutations, we multiply the number of choices at each step:

```
Total permutations = 5 × 4 × 3 = 60
```

This approach provides an intuitive understanding of permutations and how they are calculated.

## Fundamental Counting Principle

The fundamental counting principle is a concept used in combinatorics to calculate the total number of outcomes for a series of events. It states that if there are `n`

ways to perform one event and `m`

ways to perform another event, then there are `n × m`

ways to perform both events.

In the context of permutations, the fundamental counting principle can be used to calculate the total number of permutations for multiple events. For example, if we want to arrange objects from two different sets, each with `n`

and `m`

objects, the total number of permutations is `nPr × mPr`

.

## Representation of Permutation

Permutations can be represented in different ways depending on the context and requirements. Some common representations of permutations are:

**Listing the arrangements:**Permutations can be represented by listing all the possible arrangements of the objects. For example, if we have 3 objects A, B, and C, the permutations would be ABC, ACB, BAC, BCA, CAB, and CBA.**Using permutation notation:**Permutations can also be represented using permutation notation, such as`nPr`

. This notation indicates the number of permutations of`n`

objects taken`r`

at a time.

## What is Factorial?

Factorial is a mathematical operation used to calculate the product of all positive integers less than or equal to a given number. It is denoted by the symbol `!`

. For example, `5!`

(read as “5 factorial”) is calculated as `5 × 4 × 3 × 2 × 1 = 120`

.

Factorials are used in permutation calculations because they represent the number of ways objects can be arranged. The factorial of a number `n`

is denoted as `n!`

and is calculated by multiplying all positive integers less than or equal to `n`

.

## Derivation of Permutation Formula

The permutation formula can be derived using the concept of factorial. Let’s consider an example to understand the derivation:

**Example:** Suppose we have 5 objects and we want to arrange 3 of them. We can calculate the number of permutations as follows:

- For the first position, we have 5 choices.
- For the second position, we have 4 choices remaining (since we have already chosen one object).
- For the third position, we have 3 choices remaining (since we have already chosen two objects).

To find the total number of permutations, we multiply the number of choices at each position:

```
Total permutations = 5 × 4 × 3 = 60
```

This can be generalized as follows:

```
Pr = n × (n - 1) × (n - 2) × ... × (n - r + 1)
```

Simplifying this expression, we get:

```
nPr = n! / (n - r)!
```

This is the permutation formula that we use to calculate the number of permutations.

## Types of Permutation

There are different types of permutations based on the conditions and constraints applied to the arrangement of objects. Some common types of permutations are:

### Permutation with Repetition (Permutation of n Different Objects)

In this type of permutation, objects are arranged in a specific order, allowing for repetition of objects. For example, in the permutation of letters in the word “MISSISSIPPI,” each letter can be repeated, resulting in different arrangements.

### Permutation without Repetition (Permutation of n Different Objects)

In this type of permutation, objects are arranged in a specific order without any repetition. Each object can be used only once in the arrangement. For example, arranging 3 different colors (red, blue, and green) in a specific order without repetition.

### Permutation of Multi-Sets (Permutation when Objects are not Distinct)

In this type of permutation, objects are arranged when some objects are not distinct or identical. For example, arranging the letters of the word “MOM” where two letters are the same.

## Difference between Permutation and Combination

Permutation and combination are two different concepts in mathematics. While both deal with the arrangement or selection of objects, there are key differences between them.

Permutation refers to the arrangement of objects in a specific order, where the order matters. In permutation, the order of objects is important, and each arrangement is considered unique. For example, arranging the letters A, B, and C can result in different permutations such as ABC, ACB, BAC, BCA, CAB, and CBA.

On the other hand, combination refers to the selection of objects without considering the order. In combination, the order of objects is not important, and each selection is considered the same. For example, selecting three numbers from a set of numbers without considering their order.

The following table summarizes the differences between permutation and combination:

Permutation | Combination |
---|---|

Order matters | Order does not matter |

Each arrangement is unique | Each selection is considered the same |

Formula: nPr = n! / (n – r)! | Formula: nCr = n! / (r! × (n – r)!) |

Denoted by: nPr | Denoted by: nCr |

Understanding the difference between permutation and combination is important in solving various problems and selecting the appropriate method.

## Combination Probability

Combination probability is a concept used in statistics to calculate the likelihood of a specific combination occurring. It is often used in situations where the order of events or objects does not matter.

The formula to calculate combination probability is given by:

```
P(combination) = (Number of favorable outcomes) / (Total number of possible outcomes)
```

For example, if we have a bag with 5 red balls and 3 blue balls, the probability of selecting 2 red balls and 1 blue ball can be calculated using combination probability.

## Problems on Permutation

Permutation problems involve arranging objects in a specific order based on given conditions. These problems can be solved using the permutation formula and by applying the fundamental counting principle. Let’s solve a few example problems to understand the concept better.

**Example 1:** Suppose we have 4 different letters (A, B, C, and D), and we want to arrange them to form a 3-letter code. How many different codes can be formed?

**Solution:** To solve this problem, we can use the permutation formula:

```
nPr = n! / (n - r)!
```

In this case, `n = 4`

(number of letters) and `r = 3`

(number of positions in the code). Plugging these values into the formula:

```
4P3 = 4! / (4 - 3)!
= 4! / 1!
= 4 × 3 × 2 × 1 / 1
= 24 / 1
= 24
```

Therefore, there are 24 different codes that can be formed using the given letters.

**Example 2:** Suppose we have a group of 6 friends, and we want to arrange them in a row for a photograph. In how many ways can the friends be arranged?

**Solution:** To solve this problem, we can again use the permutation formula:

```
nPr = n! / (n - r)!
```

In this case, `n = 6`

(number of friends) and `r = 6`

(number of positions in the row). Plugging these values into the formula:

` Copy code````
6P6 = 6! / (6 - 6)!
= 6! / 0!
= 6 × 5 × 4 × 3 × 2 × 1 / 1
= 720 / 1
= 720
```

Therefore, there are 720 different ways to arrange the 6 friends in a row.

## Solved Examples on Permutation

Let’s solve a few more examples to further understand the concept of permutation.

**Example 1:** How many different 3-letter codes can be formed using the letters A, B, C, D, E, and F without repetition?

**Solution:** To solve this problem, we need to find the number of permutations of 6 objects taken 3 at a time:

```
6P3 = 6! / (6 - 3)!
= 6! / 3!
= (6 × 5 × 4 × 3 × 2 × 1) / (3 × 2 × 1)
= 120 / 6
= 20
```

Therefore, there are 20 different 3-letter codes that can be formed without repetition.

**Example 2:** In how many ways can the letters of the word “APPLE” be arranged?

**Solution:** To solve this problem, we need to find the number of permutations of the 5 letters in the word “APPLE”:

```
5P5 = 5! / (5 - 5)!
= 5! / 0!
= 5 × 4 × 3 × 2 × 1 / 1
= 120 / 1
= 120
```

Therefore, there are 120 different ways to arrange the letters of the word “APPLE”.

**Example 3:** In how many ways can 4 different books be arranged on a shelf?

**Solution:** To solve this problem, we need to find the number of permutations of 4 objects:

```
4P4 = 4! / (4 - 4)!
= 4! / 0!
= 4 × 3 × 2 × 1 / 1
= 24 / 1
= 24
```

Therefore, there are 24 different ways to arrange the 4 books on the shelf.

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