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A permutation is a set of things arranged in a specific sequence. Set members or elements are placed in a sequential or linear order here. For instance, the permutation of set A={1,6} is 2, as in {1,6},{6,1}. There are no alternative ways to arrange the items in set A, as you can see.
The elements in permutation must be organised in a specific sequence, whereas the order of the elements in combination is irrelevant.
Permutations definition
Basically A permutation is a method of arranging objects in a specific order. When dealing with permutation, one must think about both selection and layout. In a nutshell, ordering is critical in permutations. To put it another way, a permutation is an ordered combination.
Permutation Representation
Permutation can be represented in a variety of ways, including:
- P(n,k)
- Pnk
- nPk
- nPk
- P(n,k)
Formula
P(n,r) = n!/(n-r)! is the formula for permutation of n objects for r selection of objects.
For example, the number of ways 10 members can be awarded 3rd and 4th place is provided by:
P(10, 2) = 10!/(10-2)! = 10!/8! = (10.9.8!)/8! = 10 x 9 = 90.
Different Permutation Types
Permutation can be categorised into three categories:
n distinct objects permutation (when repetition is not allowed)
Where repetition is permitted, repetition
When the objects are not distinct, permutation is used (Permutation of multi sets)
Let’s take a closer look at each permutation situation.
n distinct objects permutation
P(n, r) indicates the number of all conceivable arrangements or permutations of n unique items taken r at a time if n is a positive integer and r is a whole number, such that r n. When using permutation without repetition, the number of options available decreases over time. It can also be written like this:
nPr
=> P(n,r) =n!/(n-r)!
nPr , indicates the “n” things to be chosen without repetition from “r” objects, where the order matters.
When letter repetition is prohibited, how many three-letter words with or without meaning may be constructed from the letters of the word SWING?
Because the word SWING comprises five letters, n = 5. Because we must frame three-letter words with or without meaning and without repetition, the total number of permutations is:
=> P(n,r) = 5!/(5-3)! = 1×2×3×4×5/ 1×2 = 60
When repetition is permitted, permutation is used.
With repetition, we can quickly determine the permutation. The exponent form can be used to write a permutation with object repetition.
When the number of objects is “n,” and the selection of objects is “r,” then
Choosing an object can be done in a variety of ways (each time).
When repetition is allowed, the permutation of items is equal to
n× n ×n……(r times)= nr
When repetition is allowed, this is the permutation formula for calculating the number of permutations possible for the selection of “r” items from the “n” objects.
Examples:
When repetition of words is allowed, how many three-letter words with or without meaning may be constructed from the letters in the word SMOKE?
Solution:
In this situation, there are five items because the word SMOKE includes five alphabets.
and r = 3 since a three-letter word must be chosen.
As a result, the permutation is:
When repetition is permitted, permutation =
53
= 125
Permutation | Combination |
Permutation refers to the process of selecting objects in which the sequence of selection is important. | The term “combination” refers to the selection of objects without regard to their order. |
To put it another way, it is the arrangement of r things selected from n objects. | To put it another way, it is the selection of r things from a set of n objects, regardless of their organisation. |
nPr= n! /(n-r)! is the permutation formula. | nCr = n!/[r!(n-r)!] Is the combination formula. |
The Basic Counting Principle
“If one operation can be performed in’m’ ways and there are n ways to conduct a second operation, then the number of ways to perform the two operations together is m x n,” according to this principle.
This idea can be extended to the case when distinct operations are carried out in m, n, p, etc.
The number of ways to perform all the operations one after the other in this case is m x n x p x… and so on.
Exercising Solutions
Example 1: How many different ways can six youngsters be placed in a line so that the two of them are always together?
(ii) They have two children who are never together.
Solution:
I Because the provided criterion requires two students to be together, we can count them as one.
As a result, the remaining 7 variations yield a total of 120.
Also, there are two ways to arrange two youngsters in a line.
As a result, the total number of arrangements is,
5! × 2! = 120 × 2 = 240 ways
(ii) There will be a total of 6 arrangements for 6 children, resulting in 720 possibilities.
We know that when two children are positioned together, they can be arranged in 240 different ways.
As a result, the total number of ways to arrange children in which two specific children are never together is 720 – 240, or 480.
Conclusion
Basically A permutation is a method of arranging objects in a specific order. When dealing with permutation, one must think about both selection and layout.
In a nutshell, ordering is critical in permutations. To put it another way, a permutation is an ordered combination.
A permutation is a set of things arranged in a specific sequence. Set members or elements are placed in a sequential or linear order here.
