Order doesn’t matter… Show
Review from Permutations (where order matters) A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can select 2 letters from that set. Each possible selection would be an example of a combination. The complete list of possible selections would be: AB, AC, and BC. Say for an example, you want to select 2 people out of 3 to send to a conference, how may ways you could select 2 people. Formula: Note that AB and BA are considered to be one combination, because the order in which objects are selected does not matter. This is the key distinction between a combination and a permutation.
The number of combinations should always be smaller than the equivalent permutations. 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. 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. Basically 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)! 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) \(\begin{array}{l}\large \Rightarrow P(n,r) = \frac{n!}{(n-r)!}\end{array} \) 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: \(\begin{array}{l} \Rightarrow P(n,r) = \frac{5!}{(5-3)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 60\end{array} \) 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: \(\begin{array}{l} \mathbf{\large \frac{n!}{p_{1}!\; p_{2}!\; p_{3}…..p_{n}!}}\end{array} \) 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 ExamplesExample 1: In how many ways 6 children can be arranged in a line, such that (i) Two particular children of them are always together (ii) Two particular children of them are never together Solution: (i) The given condition states that 2 students need to be together, hence we can consider them 1. Thus, the remaining 7 gives the arrangement in 5! ways, i.e. 120. Also, the two children in a line can be arranged in 2! Ways. Hence, the total number of arrangements will be, 5! × 2! = 120 × 2 = 240 ways (ii) The total number of arrangements of 6 children will be 6!, i.e. 720 ways. Out of the total arrangement, we know that two particular children when together can be arranged in 240 ways. Therefore, total arrangement of children in which two particular children are never together will be 720 – 240 ways, i.e. 480 ways. Example 2:Consider a set having 5 elements a,b,c,d,e. In how many ways 3 elements can be selected (without repetition) out of the total number of elements. Solution: Given X = {a,b,c,d,e} 3 are to be selected. Therefore, \(\begin{array}{l}^{5}C_{3} = 10\end{array} \) Example 3: It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible? Solution: We are given that there are 5 men and 4 women. i.e. there are 9 positions. The even positions are: 2nd, 4th, 6th and the 8th places These four places can be occupied by 4 women in P(4, 4) ways = 4! = 4 . 3. 2. 1 = 24 ways The remaining 5 positions can be occupied by 5 men in P(5, 5) = 5! = 5.4.3.2.1 = 120 ways Therefore, by the Fundamental Counting Principle, Total number of ways of seating arrangements = 24 x 120 = 2880 Practice ProblemsPractice the below listed problems:
To solve more problems or to take a test, download BYJU’S – The Learning App. Frequently Asked Questions – FAQsPermutation
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: 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!) What do you call the number of ways of selecting from a set when the order?permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor.
What term refers to the selection of objects from a set when order is not important?Permutation refers to the different ways of arranging a set of objects in a sequential order. Combination refers to several ways of choosing items from a large set of objects, such that their order does not matters.
What do you call a selection of objects from a collection where the order of selection is not taken into consideration?A combination of n objects taken r at a time is a selection which does not take into account the arrangement of the objects. That is, the order is not important.
What does it mean when order doesn't matter?If the order doesn't matter then we have a combination, if the order do matter then we have a permutation. One could say that a permutation is an ordered combination.
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