In mathematics, set theory focuses on any number of "groups" of numbers or variables, with almost no limitations. Sets even exist with absolutely no members, known as "empty sets". When referring to certain sets, however - most often those which are considered "discrete" - that is, containing a "countable" number of elements - mathematicians have developed a system of combinatorics which enables a greater understanding of sets.
The three basic combinatorial operations are: Permutation, Combination, and Enumeration.
Permutation
Permutations of a given discrete set are simply "reorderings" of the set. A perfect example of this is a typical game of Scrabble. One can imagine holding seven Scrabble tiles with various letters - H, I, J, K, E, F, B, for example. These letters may then be considered a complete set, out of which certain letters can be taken, reordered, and placed on the board to form words. These words, such as BIKE, FIB, FIE, HIKE and others, would be permutations of the original set.
A permutation can exist in any order, and does not need to include all of the members of the original set (though it cannot add any members which are not in the original set).
Permutations are very useful in many statistical cases. For example, one might ask the question: How many permutations are possible in a six-number lottery drawing? The answer to this question would then provide the odds of winning said lottery (which, needless to say, are very small indeed).
Combination
A combination of a given set is similar to a set permutation, except that it shows no concern for the order in which the set members are placed (for permutations, order is key, for, as in a game of Scrabble, FIB would be an acceptable word, while BIF is not). Rather, a combination of elements taken from a given set consists merely of an unordered sample.
For example, in the set which includes every letter in the English alphabet, all known words would be considered combinations of this set (as well as permutations), but also all made up words and random collections of letters alike.
Enumeration
Simply put, the enumeration of a set is simply a "count" of the set's elements (which is only truly possible if the original set is discrete, and therefore countable).
While on the surface, this seems very simple (the set of all letters in the English language would contain 26 members), it is also possible to find the enumeration of countable, yet infinite sets, such as the set of all natural numbers, which would be stated mathematically: -(x+1)/2, if x is odd, and x/2 if x is even.
Uses of Combinatorics
Like many other aspects of set theory, combinatorics extends far beyond mere mathematics and the ordering of numbers. As has already been described, it can be used to quantify and to conduct operations with abstract quantities like probabilities, or with finite sets such as letters in the alphabet.
In addition, combinatorics find ample use in such diverse areas of study as computer programming and combinatorial chemistry in addition to providing an insightful way of looking at many aspects of basic number theory.
References:
Combinatorics. Wolfram's Mathworld.
Combinatorics.Math Pages.
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