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A permutation is an arrangement, or listing, of objects in which the order is important. In this lesson, we will look at examples of the number of permutations of n things taken n at a time.
In another lesson, we will consider the number of permutations of n things taken r at a time.![Permute 3 1 9 X 2 Permute 3 1 9 X 2](https://www.mondialbroker.com/Immagini/Images/698a84b6-7126-406c-b2ae-7fd730bf1bd1/041c5a70-3c65-4ecb-baec-b050c3a136fb/cf0e48ea-6cc2-4540-a084-dc17708c7388.2048.jpg)
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Consider the example from the previous paragraph. The permutations were formed from 3 letters (A, B, and C), so n = 3; and each permutation consisted of 2 letters, so r = 2. For an example that counts permutations, see Sample Problem 1. Another way of looking at this question is by drawing 3 boxes. Any one of the A, B, C goes into the first box (3 ways to do this), and then the remaining one of the two letters goes into the second box (2 ways to do this), and the last remaining letter goes into the third box (only one way left to do this). Hence, total no of ways `3 × 2 × 1 = 6`.
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permutation
1.Mathsa. an ordered arrangement of the numbers, terms, etc., of a set into specified groups
b. a group formed in this way. The number of permutations of n objects taken r at a time is n&! / (n--r)&!
2. a fixed combination for selections of results on football pools
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.
Permutation
the replacement of each of the elements a of a given set by another element ϕ(a) of the same set. Each element of the initial set must be obtained precisely once as a result of the permutation. Thus, a permutation is essentially a one-to-one mapping of a set onto itself. The concept of permutation is applied chiefly to finite sets, and only this case will be considered here.
A permutation is commonly symbolized
Here, each element of the given set is written above the element corresponding to it. Since the properties of a permutation are independent of the nature of the elements a, b, …, c, the latter are usually replaced by the integers 1, 2, …, n. In the upper row, the integers are usually written in their natural order. The permutation then has the form
or simply
where ϕ1, ϕ2, …, ϕn are the numbers 1, 2, …, n, possibly in a different order. Thus, the second row of a permutation is an arrangement, ϕ1, ϕ2, …, ϕn of the numbers 1, 2, …, n. There are as many different permutations of n elements as there are arrangements—that is, n! = 1 × 2 × 3 × … × n.
The permutation
which leaves invariant all elements, is called the identity permutation. Chronicle bill management 6 7 3. Every permutation A has an inverse, that is, a permutation that carries ϕi to i. The inverse of A is denoted by A–1. For example, if
then
The result of the successive application of two permutations A and B is itself a permutation C. If A carries i to ϕi and B carries ϕi to ψi, then C carries i to ψi. C is called the product of A and B; this relationship is written C = AB. For example, if
and
then
Multiplication of permutations is not commutative; that is, in general AB ≠ BA. In the above example,
It can be easily seen that IA = AI = A, that AA–1 = A–1A = I, and that the associative law A(BC) = (AB)C holds. Thus, all the permutations of n elements form a group, which is called the symmetric group of degree n.
A permutation that interchanges two elements i and j is called a transposition and is denoted by (i, j); for example,
Any permutation can be factored into a product of transpositions. When a given permutation is factored into a product of transpositions in different ways, there will be either an even or an odd number of factors. The permutation will accordingly be said to be even or odd; for example, A = (1, 3) (5, 4) (5, 1) is an odd permutation. Define an inversion as an ordered pair of natural numbers such that the first is greater than the second. It turns out that the parity of a permutation can also be determined from the number of inversions in the lower row of the permutation if the numbers in the upper row are arranged in their natural order. The parity of the permutation coincides with the parity of the number of inversions. For example, the lower row of A contains five inversions: (3, 2), (3, 1), (2, 1), (5, 1), and (5, 4). There exist n!/2 even and n!/2 odd permutations of n elements.
A permutation that cyclically permutes a given group of elements while leaving invariant the other elements is called a cycle. The number of permuted elements is called the length of the cycle. For example, A is a cycle of length four: it carries 1 to 3, 3 to 5, 5 to 4, and 4 to 1. This fact is often denoted simply by A = (1, 3, 5, 4). A transposition is a cycle of length two. Any permutation can be factored into a product of disjoint cycles, that is, a product of cycles without common elements. For example,
REFERENCE
Kurosh, A. G. Kurs vysshei algebry, 10th ed. Moscow-Leningrad, 1971.Permute 3 1 9 X 2 3
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
permutation
[‚pər·myə′tā·shən] (mathematics)A function which rearranges a finite number of symbols; more precisely, a one-to-one function of a finite set onto itself.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
permutation
(mathematics)1. An ordering of a certain number of elements of a given set.
For instance, the permutations of (1,2,3) are (1,2,3) (2,3,1) (3,1,2) (3,2,1) (1,3,2) (2,1,3).
Permutations form one of the canonical examples of a 'group' - they can be composed and you can find an inverse permutation that reverses the action of any given permutation.
The number of permutations of r things taken from a set of n is
n P r = n! / (n-r)!
where 'n P r' is usually written with n and r as subscripts and n! is the factorial of n.
What the football pools call a 'permutation' is not a permutation but a combination - the order does not matter.
2. A bijection for which the domain and range are the same set and so
f(f'(x)) = f'(f(x)) = x.
For instance, the permutations of (1,2,3) are (1,2,3) (2,3,1) (3,1,2) (3,2,1) (1,3,2) (2,1,3).
Permutations form one of the canonical examples of a 'group' - they can be composed and you can find an inverse permutation that reverses the action of any given permutation.
The number of permutations of r things taken from a set of n is
n P r = n! / (n-r)!
where 'n P r' is usually written with n and r as subscripts and n! is the factorial of n.
What the football pools call a 'permutation' is not a permutation but a combination - the order does not matter.
2. A bijection for which the domain and range are the same set and so
f(f'(x)) = f'(f(x)) = x.
This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)
permutation
One possible combination of items out of a larger set of items. For example, with the set of numbers 1, 2 and 3, there are six possible permutations: 12, 21, 13, 31, 23 and 32.Copyright © 1981-2019 by The Computer Language Company Inc. All Rights reserved. THIS DEFINITION IS FOR PERSONAL USE ONLY. All other reproduction is strictly prohibited without permission from the publisher.
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A permutation is an arrangement, or listing, of objects in which the order is important. In this lesson, we will look at examples of the number of permutations of n things taken n at a time.
In another lesson, we will consider the number of permutations of n things taken r at a time.
Example :
Suppose we want to take a picture of three boys, Allen, Bryan and Carlos. In how many ways can the boys be arranged?
Solution:
They can be arranged in any of several ways.
Allen Bryan Carlos
Allen Carlos Bryan
Bryan Allen Carlos
Bryan Carlos Allen
Carlos Allen Bryan
Carlos Bryan Allen
Allen Carlos Bryan
Bryan Allen Carlos
Bryan Carlos Allen
Carlos Allen Bryan
Carlos Bryan Allen
There are 3 choices for the first boy, 2 choices for the second and 1 choice for the third, so the total number of permutations is 3 x 2 x 1 = 6. The 3 boys can be arranged in 6 ways.
In this example, the symbol P(3, 3) represents the number of permutations of 3 things taken 3 at a time.
P(3, 3) = 3 × 2 × 1 = 6
Example:
In how many ways can 6 people be seated in a row of 6 chairs?
We can use the permutation formula P(6, 6) which is 6 things taken 6 at a time.
P(6, 6) = 6 × 5 × 4 × 3 × 2 × 1
![Permute 3 1 9 X 2 Permute 3 1 9 X 2](https://www.mondialbroker.com/Immagini/Images/698a84b6-7126-406c-b2ae-7fd730bf1bd1/041c5a70-3c65-4ecb-baec-b050c3a136fb/cf0e48ea-6cc2-4540-a084-dc17708c7388.2048.jpg)
Six people can be seated in 720 ways.
Permute 3 1 9 X 2 1 2 In Strong Drive Sd Screws
Example:
In how many ways can seven different books be arranged on the shelf?
Solution:
We can use the permutation formula P(7, 7) which is 6 things taken 6 at a time.
P(7, 7) = 7 × 6 × 5 × 4 × 3 × 2 × 1
The books can be arranged in 5,040 ways.
1-9 Form 2020
The following video gives another example of the permutation problem.Permute 3 1 9 X 27
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