Permutation: What Is It and How to Calculate It? (Tutorial)

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Permutation: What Is It and How to Calculate It? (Tutorial) – A permutation is a mathematical term that describes the number of ways things can be ordered or arranged. With it, the order of the arrangement matters. For example, if you have three letters A, B, and C, you can arrange them in six different ways: ABC, ACB, BAC, BCA, CAB, and CBA. These are all permutations of the set {A, B, C}.

They are useful for counting and analyzing various situations and problems that involve ordering or arranging objects. For example, it can help you answer questions like:

  • How many different ways can you arrange the books on your shelf?
  • How many different passwords can you create with four digits?
  • How many different ways can you seat six people around a table?
  • How many different ways can you shuffle a deck of cards?

In this article, we will explain what permutation is, how to calculate it using formulas and examples, and what are the different types.

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What Is Permutation?

A permutation is an arrangement of objects in a definite order. Objects can be anything, such as letters, numbers, colors, shapes, cards, people, etc. The order of the objects matters in permutation. For example, the arrangement ABC is different from the arrangement BAC.

It can also be defined as a function that maps a set of objects to itself in a one-to-one manner. That is, each object in the set is assigned to exactly one other object in the same set. For example, the function f that maps {A, B, C} to {B, C, A} is a permutation.

The number of permutations of a set of n objects is denoted by n! (read as n factorial), which means the product of all positive integers from 1 to n. For example,

  • 1! = 1
  • 2! = 2 x 1 = 2
  • 3! = 3 x 2 x 1 = 6
  • 4! = 4 x 3 x 2 x 1 = 24
  • and so on.

The number of permutations of n objects is n! because there are n choices for the first object, (n-1) choices for the second object, (n-2) choices for the third object, and so on until there is only one choice for the last object.

How to Calculate Permutation?

There are two main formulas for calculating permutation: one for permutation without repetition and one for permutation with repetition.

Permutation without Repetition

Permutation without repetition means that each object in the set can be used only once in the arrangement. For example, if you have four letters A, B, C, and D, and you want to arrange them in three positions without repeating any letter, then you have:

  • 4 choices for the first position
  • 3 choices for the second position (after using one letter)
  • 2 choices for the third position (after using two letters)

Therefore, the total number of permutations without repetition is:

4 x 3 x 2 = 24

The general formula for permutation without repetition of n objects taken r at a time is:

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

where n is the total number of objects in the set and r is the number of objects to be arranged.

The formula can be derived by dividing n! by (n-r)! because (n-r)! represents the number of permutations that are not counted when we arrange r objects out of n objects. For example,

P(4,3) = 4! / (4-3)! = 24 / 1 = 24

Permutation with Repetition

Permutation with repetition means that each object in the set can be used more than once in the arrangement. For example, if you have four letters A, B, C, and D, and you want to arrange them in three positions with repetition, then you have:

  • 4 choices for the first position
  • 4 choices for the second position (even if you use the same letter as before)
  • 4 choices for the third position (even if you use the same letter as before)

Therefore, the total number of permutations with repetition is:

4 x 4 x 4 = 64

The general formula for permutation with repetition of n objects taken r at a time is:

P(n,r) = n^r

where n is the total number of objects in the set and r is the number of objects to be arranged.

The formula can be derived by multiplying n by itself r times because there are n choices for each position.

For example,

P(4,3) = 4^3 = 64

What Are the Different Types of Permutations?

There are four main types of permutations:

Simple Permutation

Simple permutation is the basic type of permutation that we have discussed so far. It is the arrangement of distinct objects in a definite order without repetition. For example, the arrangement of three letters A, B, and C in two positions is a simple permutation.

The formula for simple permutation of n objects taken r at a time is:

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

Circular Permutation

A circular permutation is the arrangement of objects in a circular order. For example, the arrangement of four people around a table is a circular permutation.

The formula for a circular permutation of n objects is:

P(n) = (n-1)!

The formula can be derived by fixing one object and arranging the remaining (n-1) objects in a simple permutation.

Permutation with Indistinguishable Objects

Permutation with indistinguishable objects is the arrangement of objects that are not distinct or identical. For example, the arrangement of four letters A, A, B, and B in three positions is a permutation with indistinguishable objects.

The formula for permutation with indistinguishable objects of n objects with k types and n1, n2,…, nk objects of each type is:

P(n,n1,n2,…,nk) = n! / (n1! x n2! x … x nk!)

The formula can be derived by dividing n! by the product of factorials of each type because each type can be arranged in its own factorial ways.

Permutation with Restrictions

Permutation with restrictions is the arrangement of objects that are subject to some conditions or limitations. For example, the arrangement of four letters A, B, C, and D in three positions such that A must be in the first position is a permutation with restrictions.

The formula for permutation with restrictions depends on the type and number of restrictions. For example,

  • If there are r fixed positions out of n positions, then the formula is:

P(n-r,r) = (n-r)! / (n-2r)!

  • If there are r forbidden positions out of n positions, then the formula is:

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

Examples

Here are some examples of how to calculate using formulas and examples.

Example 1: How many different ways can you arrange five books on a shelf?

Solution: This is a simple permutation without repetition of five objects taken five at a time. Therefore,

P(5,5) = 5! / (5-5)! = 120 / 1 = 120

There are 120 different ways to arrange five books on a shelf.

Example 2: How many different ways can you arrange six people around a circular table?

Solution: This is a circular permutation of six objects. Therefore,

P(6) = (6-1)! = 120

There are 120 different ways to arrange six people around a circular table.

Example 3: How many different ways can you arrange eight letters A, A, B, B, C, C, D, and D in four positions?

What is the permutation of 5?

The permutation of 5 is 120. This means that there are 120 ways to arrange 5 distinct objects in order.

What is the difference between permutation and combination?

A permutation is a way of arranging objects in a specific order. The order of the objects matters in a permutation. For example, the arrangement of the letters “ABCDE” is different from the arrangement of “EDCBA”.

Combination is a way of choosing objects from a group, without considering the order of the objects. The order of the objects does not matter in a combination. For example, the combination of the letters “ABC” is the same as the combination “CAB”.

Feature Permutation Combination
Order of objects Matters Does not matter
Formula n! nCr = n! / (r!(n – r)!)
Example The arrangement of the letters “ABCDE” The combination of the letters “ABC”

 

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