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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 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.
Combinatorial split-mix (split and pool) synthesis [12] [13] is based on the solid-phase synthesis developed by Merrifield. [14] If a combinatorial peptide library is synthesized using 20 amino acids (or other kinds of building blocks) the bead form solid support is divided into 20 equal portions.
This formula can be verified by counting how many times each region in the Venn diagram figure is included in the right-hand side of the formula. In this case, when removing the contributions of over-counted elements, the number of elements in the mutual intersection of the three sets has been subtracted too often, so must be added back in to ...
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.
In combinatorics, the rule of product or multiplication principle is a basic counting principle (a.k.a. the fundamental principle of counting). Stated simply, it is the intuitive idea that if there are a ways of doing something and b ways of doing another thing, then there are a · b ways of performing both actions. [1] [2]
These operations respectively form the addition and multiplication operations of a semiring on the family of (isomorphism equivalence classes of) combinatorial classes, in which the zero object is the empty combinatorial class, and the unit is the class whose only object is the empty set. [5]
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: