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Example:Based on Permutation (Arrangement of Letters)

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                                 The 6 permutations of 3 balls


In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting (rearranging in an ordered fashion) 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 anagram of a word as a permutation of its letters. The study of permutations in this sense generally belongs to the field of 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!".



In combinatorics


Permutations of multisets

In combinatorics, a permutation is usually understood to be a sequence containing each element from a finite set once, and only once. The concept of sequence is distinct from that of a set, in that the elements of a sequence appear in some order: the sequence has a first element (unless it is empty), a second element (unless its length is less than 2), and so on. In contrast, the elements in a set have no order; {1, 2, 3} and {3, 2, 1} are different ways to denote the same set. In this sense a permutation of a finite set S of n elements is equivalent to a bijection from {1, 2, ..., n} to S (in which any i is mapped to the i-th element of the sequence), or to a choice of a total ordering on S (for which x < y if x comes before y in the sequence). In this sense there are also n! Permutations of S.

There is also a weaker meaning of the term "permutation" that is sometimes used in elementary combinatorics texts, designating those sequences in which no element occurs more than once, but without the requirement to use all elements from a given set. Indeed this use often involves considering sequences of a fixed length k of elements taken from a given set of size n. These objects are also known as sequences without repetition, a term that avoids confusion with the other, more common, meanings of "permutation".

 The number of such k-permutations of n is denoted variously by such symbols as n Pk, nPk, Pn,k, or P(n,k), and its value is given by the product

which is 0 when k > n, and otherwise is equal to

(Our solved example in mathguru.com uses this concept)




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.