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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.
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 following example, due to Marshall Hall Jr., shows that the marriage condition will not guarantee the existence of a transversal in an infinite family in which infinite sets are allowed. Let F {\displaystyle {\mathcal {F}}} be the family, A 0 = N {\displaystyle A_{0}=\mathbb {N} } , A i = { i − 1 } {\displaystyle A_{i}=\{i-1\}} for i ≥ ...
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.
In mathematics, a combinatorial explosion is the rapid growth of the complexity of a problem due to the way its combinatorics depends on input, constraints and bounds. Combinatorial explosion is sometimes used to justify the intractability of certain problems.
In many cases where the principle could give an exact formula (in particular, counting prime numbers using the sieve of Eratosthenes), the formula arising does not offer useful content because the number of terms in it is excessive. If each term individually can be estimated accurately, the accumulation of errors may imply that the inclusion ...
So, a partition problem can be solved by transforming it into a distribution one and applying the correspondent operation provided by the distribution model (previous table). Following this method, we will get the number of possible ways of dividing the set. The relation between these two models is described in the following table:
Category: Theorems in combinatorics. 9 languages. ... Pages in category "Theorems in combinatorics" The following 28 pages are in this category, out of 28 total.