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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.
By the definition of the lexicographic ordering, two k-combinations that differ in their largest element c k will be ordered according to the comparison of those largest elements, from which it follows that all combinations with a fixed value of their largest element are contiguous in the list.
Two of the problems are trivial (the number of equivalence classes is 0 or 1), five problems have an answer in terms of a multiplicative formula of n and x, and the remaining five problems have an answer in terms of combinatorial functions (Stirling numbers and the partition function for a given number of parts).
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; For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation) Permutation ...
Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets S i indexed by the natural numbers , enumerative combinatorics seeks to describe a counting function which counts the number of objects in S n for each n .
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures.It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
Schematic illustration of a combinatorial species structure on five elements by using a Labelle diagram. Any species consists of individual combinatorial structures built on the elements of some finite set: for example, a combinatorial graph is a structure of edges among a given set of vertices, and the species of graphs includes all graphs on all finite sets.