Permutation and combination are fundamental concepts in mathematics that are used to count and calculate different arrangements and selections. These principles of counting have various applications in fields like probability, statistics, and computer science. Understanding the definitions, differences, formulas, and types of permutation and combination is essential for solving a wide range of mathematical problems.

## An Introduction to Permutation and Combination

Permutation and combination are mathematical techniques used to determine the number of possible outcomes in different scenarios. While they may seem similar, there is a fundamental difference between the two. Permutations are arrangements of objects or events where the order matters, while combinations are selections of objects or events where the order does not matter.

To illustrate the difference, let’s consider an example. Suppose we have three letters A, B, and C. The permutations of these letters would be ABC, ACB, BAC, BCA, CAB, and CBA. On the other hand, the combinations would be AB, AC, and BC. In permutations, the order of the elements is important, while in combinations, the order is not significant.

## What is Permutation?

Permutation, in simple terms, refers to the different ways in which a set of objects can be arranged in a specific order. In other words, permutation is all about arranging ‘r’ distinct items from a larger set of ‘n’ items, with the order of selection being crucial.

## What is Combination?

A combination, on the other hand, is a method of selecting ‘r’ items from a larger set of ‘n’ items. Unlike permutation, the order of selection doesn’t matter in combinations. Therefore, combinations are usually associated with the selection of items where the sequence is irrelevant.

## Permutation And Combination Formulas

The permutation and combination formulas play a crucial role in calculating the number of arrangements and selections. These formulas provide a systematic approach to solve various counting problems.

### Permutation Formula

The permutation formula calculates the number of permutations for arranging r objects out of n distinct objects. The formula is given by:

Where n is the total number of objects, r is the number of objects to be arranged, and “!” denotes the factorial of a number.

### Combination Formula

The combination formula determines the number of combinations for selecting r objects out of n distinct objects. The formula is given by:

```
nCr = n! / (r! * (n - r)!)
```

Where n is the total number of objects, r is the number of objects to be selected, and “!” denotes the factorial of a number.

It is important to note that the permutations are always greater than the combinations for the given values of n and r.

## Derivation of Permutation and Combination Formulas

The derivation of the permutation and combination formulas provides a deeper understanding of these counting techniques.

### Derivation of Permutations Formula

The permutations formula is derived based on the fundamental counting principle. The total number of permutations of a set of n objects taken r at a time is given by:

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

To derive this formula, let’s consider an example. Suppose we have n objects and we want to arrange them in a specific order, taking r objects at a time. The number of choices for the first object is n. Once the first object is chosen, the number of choices for the second object is reduced to n – 1. Similarly, the number of choices for the third object is n – 2, and so on. Therefore, the total number of permutations is given by:

```
P(n, r) = n * (n - 1) * (n - 2) * ... * (n - r + 1)
```

This can be written as:

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

### Derivation of Combinations Formula

The combinations formula is derived from the permutations formula. Since combinations involve selecting r objects without considering the order, we divide the number of permutations by the number of ways to arrange r objects, which is r!. The formula for combinations is given by:

```
C(n, r) = n! / (r! * (n - r)!)
```

To derive this formula, let’s consider the example of arranging n objects in a specific order. The number of permutations is given by n! / (n – r)!. However, since we are only interested in combinations, we divide this by r! to account for the multiple arrangements of the same objects. Therefore, the formula for combinations is:

```
C(n, r) = P(n, r) / r!
```

This simplifies to:

```
C(n, r) = n! / (r! * (n - r)!)
```

## How to Calculate Permutations?

Calculating permutations involves following a step-by-step process to determine the number of possible arrangements. Let’s take an example to illustrate the calculation of permutations.

Suppose we have a set of 5 numbers: 1, 2, 3, 4, and 5. We want to find the number of permutations when selecting 3 numbers from this set.

**Step 1**: Determine the values of n and r. In this case, n = 5 (total number of objects) and r = 3 (number of objects to be arranged).

**Step 2**: Apply the permutation formula. Using the permutation formula, we have: P(5, 3) = 5! / (5 – 3)! = 5! / 2! = (5 * 4 * 3 * 2 * 1) / (2 * 1) = 60

Therefore, there are 60 different permutations when selecting 3 numbers from the set of 5 numbers.

## How to Calculate Combination?

Calculating combinations involves a step-by-step process to determine the number of possible selections. Let’s use an example to illustrate the calculation of combinations.

Suppose we have a set of 6 colors: red, blue, green, yellow, orange, and purple. We want to find the number of combinations when selecting 2 colors from this set.

**Step 1**: Determine the values of n and r. In this case, n = 6 (total number of objects) and r = 2 (number of objects to be selected).

**Step 2**: Apply the combination formula. Using the combination formula, we have: C(6, 2) = 6! / (2! * (6 – 2)!) = 6! / (2! * 4!) = (6 * 5 * 4!)/(2! * 4!) = (6 * 5) / (2 * 1) = 15

Therefore, there are 15 different combinations when selecting 2 colors from the set of 6 colors.

## Properties of Permutation

Permutations have several properties that are important to understand. These properties help in solving counting problems and understanding the concept of permutations.

- Permutations can be used to calculate the number of possible arrangements of objects, events, or digits.
- The order or sequence of arrangement is important in permutations.
- The number of permutations is always greater than or equal to the number of combinations for a given set of objects.
- The number of permutations can be calculated using the permutation formula: nPr = n! / (n – r)!.
- Permutations can be used to solve problems involving seating arrangements, passwords, and word formations.

## Properties of Combination

Combinations also have certain properties that are important to understand. These properties help in solving counting problems and differentiating combinations from permutations.

- Combinations are used to calculate the number of different groups that can be formed from a set of objects or events.
- The order or sequence of arrangement is not important in combinations.
- The number of combinations is always less than or equal to the number of permutations for a given set of objects.
- The number of combinations can be calculated using the combination formula: nCr = n! / (r! * (n – r)!).
- Combinations can be used to solve problems involving selections, committees, and groups.

## Relation Between Permutation and Combination

There is a close relationship between permutation and combination. They are related through the formula:

```
nCr = nPr / r!
```

This formula shows that the number of combinations is equal to the number of permutations divided by the factorial of r. This relationship can be used to convert between permutations and combinations.

For example, if we have the number of permutations (nPr), we can calculate the number of combinations (nCr) by dividing it by r!.

## The Fundamental Principle of Addition

The fundamental principle of addition is a counting principle used to calculate the total number of possibilities when two or more events are combined. It states that if event A can occur in m ways and event B can occur in n ways, then the total number of possibilities is m + n.

For example, if we have two sets of objects, A and B, with m and n objects respectively, the total number of ways to arrange or select objects from both sets is m + n.

This principle is used in many counting problems involving permutations and combinations, where different events or objects are combined to determine the total number of outcomes.

## The Fundamental Principle of Multiplication

The fundamental principle of multiplication is another counting principle used to calculate the total number of possibilities when two or more events occur together. It states that if event A can occur in m ways and event B can occur in n ways, then the total number of possibilities is m * n.

For example, if we have two sets of objects, A and B, with m and n objects respectively, the total number of ways to arrange or select objects from both sets is m * n.

This principle is used in many counting problems involving permutations and combinations, where different events or objects occur together to determine the total number of outcomes.

## What is Factorial?

Factorial is a mathematical operation denoted by the symbol “!”. It is used to calculate the product of all positive integers from 1 to a given number. The factorial of a number n is represented as n!.

For example, the factorial of 4, denoted as 4!, is calculated as:

4! = 4 * 3 * 2 * 1 = 24

Factorials are used in permutation and combination formulas to calculate the total number of arrangements or selections. They are also used in probability calculations and in solving various counting problems.

## Counting Formulas for Permutations and Combinations

The counting formulas for permutations and combinations are essential tools in solving counting problems. These formulas provide a systematic approach to calculate the number of possible arrangements and selections.

The permutation formula, nPr, calculates the number of permutations for arranging r objects out of n distinct objects. It is given by:

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

The combination formula, nCr, determines the number of combinations for selecting r objects out of n distinct objects. It is given by:

```
nCr = n! / (r! * (n - r)!)
```

These formulas can be applied to various counting problems, such as arranging objects on a shelf, selecting a committee, or forming a password.

## Types of Permutation

Permutations can be classified into different types based on the characteristics of the objects being arranged. Understanding these types is essential for solving counting problems that involve specific conditions or restrictions.

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

Permutation with repetition involves arranging objects where some objects are repeated. In this type of permutation, the number of arrangements is calculated using the formula:

```
nPr = n! / (p1! * p2! * ... * pk!)
```

Where n is the total number of objects, and p1, p2, …, pk are the number of objects of each distinct kind.

For example, if we have the letters A, B, B, C, and C, the number of different arrangements would be:

```
5! / (1! * 2! * 2!) = 30
```

So, there are 30 different arrangements of the letters A, B, B, C, and C.

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

Permutation without repetition involves arranging objects where each object is unique and not repeated. In this type of permutation, the number of arrangements is calculated using the formula:

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

Where n is the total number of objects, and r is the number of objects to be arranged.

For example, if we have the numbers 1, 2, 3, and we want to arrange them in a specific order, the number of permutations would be:

` Copy code````
3! / (3 - r)! = 6
```

So, there are 6 different arrangements of the numbers 1, 2, 3.

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

Permutation of multi-sets involves arranging objects where some objects are identical. In this type of permutation, the number of arrangements is calculated using the formula:

```
nPr = n! / (p1! * p2! * ... * pk!)
```

Where n is the total number of objects, and p1, p2, …, pk are the number of objects of each distinct kind.

For example, if we have the letters A, B, B, C, and C, the number of different arrangements would be:

```
5! / (1! * 2! * 2!) = 30
```

So, there are 30 different arrangements of the letters A, B, B, C, and C.

## Difference between Permutation and Combination

While permutation and combination are related concepts, there are several key differences between them. Understanding these differences is crucial in choosing the appropriate counting technique for different scenarios.

- Permutations are used when the order or sequence of arrangement is important, while combinations are used when only the number of possible groups is required, and the order does not matter.
- Permutations are used for objects of different kinds, while combinations are used for objects of the same kind.
- The number of permutations is always greater than the number of combinations for a given set of objects.
- The formula for calculating permutations is nPr = n! / (n – r)!, while the formula for calculating combinations is nCr = n! / (r! * (n – r)!).
- Permutations are used in problems involving seating arrangements, word formations, and password generation, while combinations are used in problems involving selections, committees, and groups.

The table below summarizes the difference between permutation and combination:

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

Order matters | Order does not matter |

Objects are of different kinds | Objects are of the same kind |

nPr = n! / (n – r)! | nCr = n! / (r! * (n – r)!) |

Used in seating arrangements, word formations, and password generation | Used in selections, committees, and groups |

Permutations are greater in number | Combinations are lesser in number |

## Combination Probability

Combination probability involves calculating the probability of selecting a specific combination from a set of objects or events. It is used in probability theory to determine the likelihood of a particular combination occurring.

The probability of selecting a combination can be calculated using the formula:

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

For example, suppose we have a bag containing 10 marbles, numbered from 1 to 10. If we want to find the probability of selecting 3 marbles with numbers 2, 5, and 8, the calculation would be:

`P(combination) = 1 / (Total number of possible outcomes) `

Since there is only one favorable outcome (selecting marbles with numbers 2, 5, and 8) and the total number of possible outcomes is the number of combinations of selecting 3 marbles from 10, the probability can be calculated.

Combination probability is used in various real-life scenarios, such as drawing cards from a deck, selecting lottery numbers, or choosing a specific combination of items from a set.

## Using Permutation vs Combination to Solve Probability Problems

Permutation and combination are powerful tools in solving probability problems. They allow us to calculate the number of possible outcomes and determine the likelihood of a specific event occurring.

When solving probability problems, it is important to differentiate between permutation and combination. If the order or arrangement is important, we use permutation. If the order does not matter, we use combination.

For example, suppose we have a deck of cards and we want to find the probability of drawing a specific hand. If the order of the cards in the hand is important, we use permutation. If the order does not matter, we use combination.

By applying the appropriate counting technique and probability formula, we can calculate the probability of different events or combinations occurring.

## Solved Examples on Permutation and Combination

Let’s solve some examples to further understand how permutation and combination are applied in practice.

**Example 1**: Patricia has to choose 2 dresses from 12 in her wardrobe. In how many ways can she choose them?

**Solution**: Patricia has to choose 2 dresses out of 12. The order doesn’t matter here. Thus, combinations are used here. She can choose them in 12C2 ways. 12C2 = 12! / (2! * (12 – 2)!) = 12! / (2! * 10!) = (12 * 11) / (2 * 1) = 66

**Answer**: Therefore, there are 66 ways.

**Example 2:** How many ways can a committee of 4 members be selected from a group of 10 individuals?

**Solution**: As the problem involves selecting members and the order does not matter, we use the combination formula.

nCr = n! / [r!(n-r)!] = 10C4 = 10! / [4!(10-4)!] = 210 ways.

**Example 3:** How many different 4-digit numbers can be formed from the digits 1, 2, 3, 4, 5 if repetition of digits is not allowed?

**Solution**: As the problem involves arranging digits and the order matters, we use the permutation formula.

nPr = n! / (n-r)! = 5P4 = 5! / (5-4)! = 120 ways.

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