Permutation: What is it and how to calculate it?

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Permutation is a mathematical concept that deals with the arrangement of objects in a specific order. It is often used to study the possible outcomes of an event, such as a lottery, a password, or a seating plan.

In this article, we will explain what permutation is, how to calculate it, and how it differs from another concept called combination. We will also provide some examples and applications of permutation in various fields of science and logic.

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What is Permutation?

A permutation of a set of objects is a way of ordering them in a sequence, such that each object appears exactly once. For example, if we have three objects A, B, and C, there are six possible permutations of them:

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  • ABC
  • ACB
  • BAC
  • BCA
  • CAB
  • CBA

The order of the objects matters in a permutation. For instance, ABC and ACB are different permutations, even though they have the same objects. Permutations are also called ordered arrangements or ordered selections.

How to Calculate Permutation?

To calculate the number of permutations of n distinct objects, we can use the factorial notation. The factorial of a positive integer n, denoted by n! is the product of all positive integers from 1 to n. For example,

  • 1! = 1
  • 2! = 1 x 2 = 2
  • 3! = 1 x 2 x 3 = 6
  • 4! = 1 x 2 x 3 x 4 = 24

The number of permutations of n distinct objects is equal to n! This is because we have n choices for the first object, n – 1 choices for the second object, n – 2 choices for the third object, and so on until we have only one choice for the last object. For example,

  • The number of permutations of 3 distinct objects is 3! = 6.
  • The number of permutations of 4 distinct objects is 4! = 24.
  • The number of permutations of 5 distinct objects is 5! = 120.

Sometimes, we may want to find the number of permutations of r objects selected from a set of n distinct objects, where r < n. This means that we do not use all the objects in the set, but only some of them. For example, if we have five letters A, B, C, D, and E, and we want to find the number of permutations of three letters selected from them.

To calculate the number of permutations of r objects selected from n distinct objects, we can use the following formula:

nPr = n! / (n – r)!

This formula can be derived from the factorial formula by dividing by (n – r)! This is because we do not care about the order of the remaining (n – r) objects that are not selected. For example,

  • The number of permutations of 3 letters selected from 5 letters is:

5P3 = 5! / (5 – 3)! = 120 / 2 = 60

  • The number of permutations of 2 digits selected from 10 digits (0 to 9) is:

10P2 = 10! / (10 – 2)! = 3628800 / 40320 = 90

Permutations can also be represented by using a notation called P(n,r), where n and r are the same as above. For example,

  • P(5,3) = 5P3 = 60
  • P(10,2) = 10P2 = 90

How does Permutation differ from Combination?

Permutations are related to another mathematical concept called combination. A combination is a way of selecting objects from a set without regard to their order. For example, if we have three objects A, B, and C, there are only three possible combinations of them:

  • AB
  • AC
  • BC

The order of the objects does not matter in a combination. For instance, AB and BA are the same combination, even though they have different orders. Combinations are also called unordered arrangements or unordered selections.

To calculate the number of combinations of r objects selected from n distinct objects, we can use the following formula:

nCr = n! / [r! x (n – r)!]

This formula can be derived from the permutation formula by dividing by r!. This is because we do not care about the order of the r objects that are selected. For example,

  • The number of combinations of 3 letters selected from 5 letters is:

5C3 = 5! / [3! x (5 – 3)!] = 120 / [6 x 2] = 10

  • The number of combinations of 2 digits selected from 10 digits (0 to 9) is:

10C2 = 10! / [2! x (10 – 2)!] = 3628800 / [2 x 40320] = 45

Combinations can also be represented by using a notation called C(n,r), where n and r are the same as above. For example,

  • C(5,3) = 5C3 = 10
  • C(10,2) = 10C2 = 45

The main difference between permutations and combinations is that permutations care about the order of the objects, while combinations do not. Therefore, the number of permutations is always greater than or equal to the number of combinations, for the same values of n and r.

Why is Permutation important?

Permutations are useful for counting the number of ways an event can occur or a task can be performed. For example,

  • If there are six candidates for a job interview, and only three will be selected for the final round, how many ways can the final round be arranged?

The answer is P(6,3) = 6P3 = 120.

  • If there are four books on a shelf, and we want to arrange them in different orders, how many ways can we do that?

The answer is P(4,4) = 4P4 = 4! = 24.

Permutations can also be used to create codes or passwords that are hard to guess. For example,

  • If a password consists of four letters chosen from A to Z (26 letters), how many possible passwords are there?

The answer is P(26,4) = 26P4 = 358800.

  • If a license plate consists of three letters followed by three digits (0 to 9), how many possible license plates are there?

The answer is P(26,3) x P(10,3) = (26P3) x (10P3) = (15600) x (720) =11232000.

Permutations are also related to probability and statistics. They help us to calculate the likelihood of an event happening or not happening, based on the number of possible outcomes and the number of favorable outcomes. For example,

  • If we toss a coin three times, what is the probability of getting exactly two heads?

The answer is P(3,2) / 2^3 = (3P2) / (2^3) = (6) / (8) = 0.75.

  • If we draw five cards from a standard deck of 52 cards, what is the probability of getting a royal flush (A, K, Q, J, and 10 of the same suit)?

The answer is P(4,1) x P(5,5) / P(52,5) = (4P1) x (5P5) / (52P5) = (4) x (120) / (311875200) = 0.00000154.

In conclusion, Permutation is a mathematical concept that deals with the arrangement of objects in a specific order. It is often used to study the possible outcomes of an event, such as a lottery, a password, or a seating plan.

To calculate the number of permutations of n distinct objects, we can use the factorial notation n!. To calculate the number of permutations of r objects selected from n distinct objects, we can use the formula nPr = n! / (n – r)!.

Permutation differs from combination in that permutation cares about the order of the objects, while combination does not. To calculate the number of combinations of r objects selected from n distinct objects, we can use the formula nCr = n! / [r! x (n – r)!].

Permutations are important for counting the number of ways an event can occur or a task can be performed. They are also useful for creating codes or passwords that are hard to guess. They are also related to probability and statistics, as they help us to calculate the likelihood of an event happening or not happening.

Permutation: Frequently Asked Questions (F&Qs)

Permutation

What is the permutation of 5?

The number of permutations of 5 objects is calculated as follows:

P(5, 5) = 5! / (5 – 5)! = 120

Here, n = 5 represents the number of objects and r = 5 represents the number of objects to be selected. The formula for calculating permutations is given by:

P(n, r) = n! / (n – r)!

Where n is the total number of objects and r is the number of objects to be selected. The exclamation mark (!) denotes the factorial function.

What are the 4 types of permutations?

The four types of permutations are:

  • Permutations with repetition: These are the permutations where the same object can be chosen more than once. For example, the number of ways to form a four-digit PIN code with digits from 0 to 9 is 10^4, since each digit can be repeated.
  • Permutations without repetition: These are the permutations where the same object can be chosen only once. For example, the number of ways to arrange four different books on a shelf is 4!, since each book can be placed only once.
  • Permutations with multi-sets: These are the permutations where some objects are identical or indistinguishable. For example, the number of ways to arrange the letters of the word “BING” is 4!/2!, since there are two identical letters “B”.
  • Circular permutations: These are the permutations where the objects are arranged in a circular order. For example, the number of ways to seat four people around a round table is (4-1)!, since the relative order of the people matters, not their absolute position.

What is the permutation of 8p8?

The permutation of 8p8 is the number of ways to arrange 8 distinct objects in a specific order. The formula for calculating the permutation of 8p8 is:

where 8! means 8 factorial, which is the product of all positive integers less than or equal to 8. For example, 8! = 8 x 7 x 6 x … x 1.

Using this formula, we can find the value of the permutation of 8p8 by simplifying the expression:

Therefore, the permutation of 8p8 is 40320. This means that there are 40320 different ways to order 8 distinct objects.

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