# Permutation vs. Combination: What's the Difference?

Edited by Aimie Carlson || By Harlon Moss || Updated on October 15, 2023
Permutation is the arrangement of items in a specific order, while combination is the selection of items without considering order.

## Key Differences

Permutation refers to the arrangement of items in a specific sequence or order. This means the order in which the items are arranged is important. For example, the permutations of the letters A, B, and C are ABC, ACB, BAC, BCA, CAB, and CBA. Each arrangement is considered unique in permutation.
Combination, on the other hand, deals with the selection of items without considering their order. Here, it's the group of items that matters, not the sequence in which they appear. For the same set of letters, A, B, and C, the combinations are ABC, AB, AC, and BC. Note that AC and CA are the same combination because order does not matter.
It's easier to think of permutation as a specific sequence. For example, consider a race. The order in which the participants finish (first, second, third, etc.) is a permutation. In contrast, combination could be likened to a team selection, where the members of the team are important, but their sequence is not.
Both permutation and combination are foundational concepts in the field of combinatorics, a branch of mathematics that deals with counting and arrangements. While permutations focus on sequences and order, combinations are all about groups and selection.

## Comparison Chart

### Basic Definition

Order matters.
Order doesn't matter.

### Example (ABC)

ABC, ACB, BAC, BCA, CAB, CBA.
ABC, AB, AC, BC.

### Grammar Usage

"The permutation of words in the sentence changes its meaning."
"The combination of colors in the painting is stunning."

### In Mathematics

NPr = n! / (n-r)!
NCr = n! / (r! * (n-r)!)

### Real-Life Scenario

Arranging books on a shelf.
Selecting books to read from a shelf.

## Permutation and Combination Definitions

#### Permutation

Permutation is the act of altering the order of a set.
The music playlist has several permutations, depending on the mood.

#### Combination

Combination represents the total ways items can be grouped together.
There are 3 combinations for the colors red, blue, and green taken two at a time.

#### Permutation

Permutation signifies a distinct and unique arrangement within a set.
Each permutation of the puzzle pieces results in a different picture.

#### Combination

Combination deals with selection without focusing on sequencing.
The combination of notes creates a beautiful chord in music.

#### Permutation

Permutation denotes the total number of ways items can be sequenced.
There are 6 permutations for the three colors red, blue, and green.

#### Combination

Combination is the act of choosing elements from a larger set.
The combination of ingredients makes the dish unique.

#### Permutation

Permutation is the arrangement of objects in a specific order.
The permutation of the numbers 1, 2, and 3 are 123, 132, 213, 231, 312, and 321.

#### Combination

Combination is the selection of items without regard for order.
The combination of fruits for the salad includes apples, bananas, and oranges.

#### Permutation

Permutation involves both selection and arrangement of items.
The permutation of letters in a word game can lead to various outputs.

#### Combination

Combination signifies a collection or assortment of items.
Each combination of team members brings different strengths.

#### Permutation

The process of altering the order of a given set of objects in a group.

#### Combination

The act of combining or the state of being combined.

#### Permutation

The result of such a process; a rearrangement or recombination of elements
Permutations of gene order.

#### Combination

The result of combining.

## FAQs

#### How is combination different from permutation?

In combination, order doesn't matter, while in permutation, order is important.

#### Can the terms permutation and combination be used interchangeably?

No, because they refer to different mathematical concepts.

#### How many permutations are there for a set of n items taken r at a time?

There are nPr = n! / (n-r)! permutations.

#### How many combinations are there for a set of n items taken r at a time?

There are nCr = n! / (r! * (n-r)!) combinations.

#### Why is the order important in permutation?

Because permutation considers each unique arrangement as distinct.

#### Can a set have the same permutation and combination value?

Yes, when the set is taken all at once, its permutation and combination values are both 1.

#### Which is more complex to calculate, permutation or combination?

Permutation is generally more complex due to the importance of order.

#### Why is there no consideration of order in combinations?

Because combinations focus on groupings, not sequences.

#### Does changing the order of items affect combination?

No, order doesn't matter in combinations.

#### What is the significance of permutation in real life?

It can represent scenarios like race outcomes, password arrangements, and more.

#### Is permutation always greater than combination for the same set?

Yes, because permutation considers different orders, while combination does not.

#### Can a single item have permutations?

Yes, but there's only one permutation for a single item.

#### How are permutation and combination used in statistics?

They help in calculating probabilities and outcomes in various scenarios.

#### Are permutation and combination concepts only applicable in mathematics?

No, they are used in various fields like computer science, statistics, and everyday life scenarios.

#### How is combination used in real-life scenarios?

Combination is used in scenarios like team selections, food recipes, and more.

#### What does "n!" mean in permutation and combination formulas?

"n!" means n factorial, which is the product of all positive integers up to n.

#### Are there any real-life scenarios where both permutation and combination apply?

Yes, such as selecting and arranging team members for specific positions.

#### What is a permutation?

A permutation is the arrangement of objects in a specific order.

#### Is every permutation also a combination?

Yes, but not every combination is a permutation.

#### How does changing the order of items affect permutation?

Changing order creates a new permutation.