Document Sample
Matrix Powered By Docstoc

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 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!".

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 algebra
and particularly in 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 symmetric group. The key to its structure is the possibility to compose
permutations: performing two given rearrangements in succession defines a third rearrangement.
                                                  Know More About :- Types of Quadrilaterals                                                               Page : 1/3
In elementary combinatorics, the name "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).

Combination:-In mathematics a combination is a way of selecting several things out of a larger group,
where (unlike permutations) order does not matter. In smaller cases it is possible to count the number of
combinations. For example given three fruit, say an apple, orange and pear, there are three
combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a
pear and an orange. More formally a k-combination of a set S is a subset of k distinct elements of S. If
the set has n elements the number of k-combinations is equal to the binomial coefficient which can be
written using factorials as whenever , and which is zero when .

Combinations can refer to the combination of n things taken k at a time without or with repetitions.[1]
In the above example repetitions were not allowed. If however it was possible to have two of any one
kind of fruit there would be 3 more combinations: one with two apples, one with two oranges, and one
with two pears.With large sets, it becomes necessary to use more sophisticated mathematics to find the
number of combinations. For example, a poker hand can be described as a 5-combination (k = 5) of
cards from a 52 card deck (n = 52). The 5 cards of the hand are all distinct, and the order of cards in the
hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand
at random is 1 / 2,598,960.A k-combination with repetitions, or k-multicombination, or multiset of size
k from a set S is given by a sequence of k not necessarily distinct elements of S, where order is not
taken into account: two sequences of which one can be obtained from the other by permuting the terms
define the same multiset.

In other words, the number of ways to sample k elements from a set of n elements allowing for
duplicates (i.e., with replacement) but disregarding different orderings (e.g. {2,1,2} = {1,2,2}).If S has
n elements, the number of such k-multicombinations is also given by a binomial coefficient, namely
byThere is an easy way to understand the above result. Label the elements of S with numbers 0, 1, ..., n
− 1, and choose a k-combination from the set of numbers { 1, 2, ..., n + k − 1 } (so that there are n − 1
unchosen numbers).

                                             Read More About :- Properties of Parallelograms                                                             Page : 2/3
     Thank You


Shared By: