A permutation is an arrangement of objects in a definite order. The members or elements of sets are arranged here in a sequence or linear order. For example, the permutation of set A={1,6} is 2, such as {1,6}, {6,1}. As you can see, there are no other ways to arrange the elements of set A. Show
In permutation, the elements should be arranged in a particular order whereas in combination the order of elements does not matter. Also, read: Permutation And Combination When we look at the schedules of trains, buses and the flights we really wonder how they are scheduled according to the public’s convenience. Of course, the permutation is very much helpful to prepare the schedules on departure and arrival of these. Also, when we come across licence plates of vehicles which consists of few alphabets and digits. We can easily prepare these codes using permutations. Definition of PermutationBasically Permutation is an arrangement of objects in a particular way or order. While dealing with permutation one should concern about the selection as well as arrangement. In Short, ordering is very much essential in permutations. In other words, the permutation is considered as an ordered combination. Representation of PermutationWe can represent permutation in many ways, such as: FormulaThe formula for permutation of n objects for r selection of objects is given by: P(n,r) = n!/(n-r)! For example, the number of ways 3rd and 4th position can be awarded to 10 members is given by: P(10, 2) = 10!/(10-2)! = 10!/8! = (10.9.8!)/8! = 10 x 9 = 90 Click here to understand the method of calculation of factorial. Types of PermutationPermutation can be classified in three different categories:
Let us understand all the cases of permutation in details. Permutation of n different objectsIf n is a positive integer and r is a whole number, such that r < n, then P(n, r) represents the number of all possible arrangements or permutations of n distinct objects taken r at a time. In the case of permutation without repetition, the number of available choices will be reduced each time. It can also be represented as: nPr. P(n, r) = n(n-1)(n-2)(n-3)……..upto r factors P(n, r) = n(n-1)(n-2)(n-3)……..(n – r +1) Here, “nPr” represents the “n” objects to be selected from “r” objects without repetition, in which the order matters. Example: How many 3 letter words with or without meaning can be formed out of the letters of the word SWING when repetition of letters is not allowed? Solution: Here n = 5, as the word SWING has 5 letters. Since we have to frame 3 letter words with or without meaning and without repetition, therefore total permutations possible are: Permutation when repetition is allowedWe can easily calculate the permutation with repetition. The permutation with repetition of objects can be written using the exponent form. When the number of object is “n,” and we have “r” to be the selection of object, then; Choosing an object can be in n different ways (each time). Thus, the permutation of objects when repetition is allowed will be equal to, n × n × n × ……(r times) = nr This is the permutation formula to compute the number of permutations feasible for the choice of “r” items from the “n” objects when repetition is allowed. Example: How many 3 letter words with or without meaning can be formed out of the letters of the word SMOKE when repetition of words is allowed? Solution: The number of objects, in this case, is 5, as the word SMOKE has 5 alphabets. and r = 3, as 3-letter word has to be chosen. Thus, the permutation will be: Permutation (when repetition is allowed) = 53 = 125 Permutation of multi-setsPermutation of n different objects when P1 objects among ‘n’ objects are similar, P2 objects of the second kind are similar, P3 objects of the third kind are similar ……… and so on, Pk objects of the kth kind are similar and the remaining of all are of a different kind, Thus it forms a multiset, where the permutation is given as: Difference Between Permutation and CombinationThe major difference between the permutation and combination are given below:
Fundamental Counting PrincipleAccording to this principle, “If one operation can be performed in ‘m’ ways and there are n ways of performing a second operation, then the number of ways of performing the two operations together is m x n “. This principle can be extended to the case in which the different operation be performed in m, n, p, . . . . . . ways. In this case the number of ways of performing all the operations one after the other is m x n x p x . . . . . . . . and so on Read More:
Video LessonsPermutation and CombinationProblems based on PermutationsSolved Examples
Practice ProblemsPractice the below listed problems:
To solve more problems or to take a test, download BYJU’S – The Learning App. Permutation is a way of changing or arranging the elements or objects in a linear order. The formula for permutation for n objects taken r at a time is given by: The permutation of an arrangement of objects or elements in order, depends on three conditions: When repetition of elements is not allowed When repetition of elements is allowed When the elements of a set are not distinct Let n be the number of objects and r be the selection of objects, then if repetition is allowed, the permutation of objects will be n × n × n × ……(r times) = n^r The permutation formula for multisets where all the elements are not distinct is given by: n!/(P1!P2!…Pn!) |