|
Permutations And Combinations
Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: * Twelvefold way Explained separately in a more accessible way: * Combination * Permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ... For meanings outside of mathematics, please see both words’ disambiguation pages: * Combination (other) * Permutation (other) {{Mathematics disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twelvefold Way
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number. The idea of the classification is credited to Gian-Carlo Rota, and the name was suggested by Joel Spencer. Overview Let and be finite sets. Let n=, N, and x=, X, be the cardinalities of the sets. Thus is a set with elements, and is a set with elements. The general problem we consider is the enumeration of equivalence classes of functions f: N \to X. The functions are subject to one of the three following restrictions: * No condition: each in may be sent by to any in , and each may occur multiple times. * is injective: each value f(a) for in must be distinct from every other, and so each in may occur at most once in the image of . * is surjective: for each in there must be at least one in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combination
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a 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''. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not matter.) If the set has ''n'' elements, the number of ''k''-combinations, denoted by C(n,k) or C^n_k, is equal to the binomial coefficient \binom nk = \frac, which can be written using factorials as \textstyle\frac whenever k\leq n, and which is zero when k>n. This formula can be derived from the fact that each ''k''-combination of a set ''S'' of ''n'' members has k! permu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combination (other)
A combination is a mathematical collection of things in a context where their specific order is irrelevant. Combination, combinations, or combo may also refer to: * Combination (chess), a relatively long sequence of chess moves, involving temporary loss of materials * Combination (jump), in horseback riding * Combination bus, a purpose-built truck with a "passenger container" * Combination company, in the late 19th century, a touring theater company that performed only one play * Combination Game, a style of association football based around teamwork and cooperation * Combination meal, typically includes food items and a beverage * Combination tone, a psychoacoustic phenomenon * Combinations (finance), the simultaneous buying or selling of one or more options that differ in one or more of the options' variables * ''Combinations'' (album), 2007 album by Eisley * Striking combination, several strikes in a row in combat sports * ''The Combination'' (film), 2009 Australian drama ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
