Permutations and Combinations

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					                   Permutations and Combinations

Permutations and Combinations
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 :- Associative Property of Multiplication                                                               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 . The set of all k-
combinations of a set S is sometimes denoted by .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.

Number of k-combinations:-The number of k-combinations from a given set S of n elements is often
denoted in elementary combinatorics texts by C(n, k), or by a variation such as , , or even (the latter
form is standard in French, Russian, and Polish texts[citation needed]). The same number however
occurs in many other mathematical contexts, where it is denoted by (often read as "n choose k");
notably it occurs as coefficient in the binomial formula, hence its name binomial coefficient. One can
define for all natural numbers k at once by the relation To see that these coefficients count k-
combinations from S, one can first consider a collection of n distinct variables Xs labeled by the
elements s of S, and expand the product over all elements of S:

                                                  Read More About :- How to Do Long Division                                                            Page : 2/3
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