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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 Math.Edurite.com 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 Math.Edurite.com Page : 2/3 Thank You Math.Edurite.Com

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posted: | 7/7/2012 |

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