Changes

==Origin==

==Origin==

[https://nordan.daynal.org/wiki/index.php?title=English#ca._1100-1500_.09THE_MIDDLE_ENGLISH_PERIOD Middle English] ''permutacioun'' [[exchange]], [[transformation]], from Anglo-French, from [[Latin]] ''permutation''-, ''permutatio'', from ''permutare''

[https://nordan.daynal.org/wiki/index.php?title=English#ca._1100-1500_.09THE_MIDDLE_ENGLISH_PERIOD Middle English] ''permutacioun'' [[exchange]], [[transformation]], from Anglo-French, from [[Latin]] ''permutation''-, ''permutatio'', from ''permutare''

*[http://en.wikipedia.org/wiki/14th_century 14th Century]

*[https://en.wikipedia.org/wiki/14th_century 14th Century]

==Definitions==

==Definitions==

*1: often major or fundamental [[change]] (as in [[character]] or condition) based primarily on rearrangement of existent elements <the system has gone through several permutations>; also : a [[form]] or variety resulting from such [[change]] <[[technology]] available in various permutations>

*1: often major or fundamental [[change]] (as in [[character]] or condition) based primarily on rearrangement of existent elements <the system has gone through several permutations>; also : a [[form]] or variety resulting from such [[change]] <[[technology]] available in various permutations>

*2a : the [[act]] or [[process]] of changing the lineal order of an ordered set of objects  

*2a : the [[act]] or [[process]] of changing the lineal order of an ordered set of objects  

==Description==

==Description==

In [[mathematics]], the notion of '''permutation''' is used with several slightly different [[meanings]], all related to the act of permuting (rearranging) objects or [[values]]. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). One might define an [http://en.wikipedia.org/wiki/Anagram anagram] of a word as a permutation of its [[letters]]. The [[study]] of permutations in this sense generally belongs to the field of [http://en.wikipedia.org/wiki/Combinatorics combinatorics].

In [[mathematics]], the notion of '''permutation''' is used with several slightly different [[meanings]], all related to the act of permuting (rearranging) objects or [[values]]. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). One might define an [https://en.wikipedia.org/wiki/Anagram anagram] of a word as a permutation of its [[letters]]. The [[study]] of permutations in this sense generally belongs to the field of [https://en.wikipedia.org/wiki/Combinatorics combinatorics].

The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×...×2×1, which number is called "n factorial" and written "n!".

The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×...×2×1, which number is called "n factorial" and written "n!".

Permutations occur, in more or less prominent ways, in almost every domain of [[mathematics]]. They often arise when different orderings on certain [[finite]] sets are considered, possibly only because one wants to ignore such orderings and needs to know how many [[configurations]] are thus identified. For similar reasons permutations arise in the study of sorting [[algorithms]] in [[computer science]].

Permutations occur, in more or less prominent ways, in almost every domain of [[mathematics]]. They often arise when different orderings on certain [[finite]] sets are considered, possibly only because one wants to ignore such orderings and needs to know how many [[configurations]] are thus identified. For similar reasons permutations arise in the study of sorting [[algorithms]] in [[computer science]].

In [http://en.wikipedia.org/wiki/Algebra algebra] and particularly in [http://en.wikipedia.org/wiki/Group_theory group theory], a permutation of a set S is defined as a bijection from S to itself (i.e., a map S → S for which every element of S occurs exactly once as image value). This is related to the rearrangement of S in which each element s takes the place of the corresponding f(s). The collection of such permutations form a [http://en.wikipedia.org/wiki/Symmetric_group symmetric group]. The key to its [[structure]] is the possibility to [[compose]] permutations: performing two given rearrangements in succession defines a third rearrangement, the composition. Permutations may act on composite objects by rearranging their components, or by certain replacements ([[substitution]]s) of [[symbols]].

In [https://en.wikipedia.org/wiki/Algebra algebra] and particularly in [https://en.wikipedia.org/wiki/Group_theory group theory], a permutation of a set S is defined as a bijection from S to itself (i.e., a map S → S for which every element of S occurs exactly once as image value). This is related to the rearrangement of S in which each element s takes the place of the corresponding f(s). The collection of such permutations form a [https://en.wikipedia.org/wiki/Symmetric_group symmetric group]. The key to its [[structure]] is the possibility to [[compose]] permutations: performing two given rearrangements in succession defines a third rearrangement, the composition. Permutations may act on composite objects by rearranging their components, or by certain replacements ([[substitution]]s) of [[symbols]].

In elementary combinatorics, the name "[http://en.wikipedia.org/wiki/Permutations_and_combinations permutations and combinations]" refers to two related [[problems]], both counting possibilities to select k distinct elements from a set of n elements, where for k-permutations the order of selection is taken into account, but for k-combinations it is ignored. However k-permutations do not correspond to permutations as discussed in this article (unless k = n).[http://en.wikipedia.org/wiki/Permutation]

In elementary combinatorics, the name "[https://en.wikipedia.org/wiki/Permutations_and_combinations permutations and combinations]" refers to two related [[problems]], both counting possibilities to select k distinct elements from a set of n elements, where for k-permutations the order of selection is taken into account, but for k-combinations it is ignored. However k-permutations do not correspond to permutations as discussed in this article (unless k = n).[https://en.wikipedia.org/wiki/Permutation]

[[Category: Mathematics]]

[[Category: Mathematics]]