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The rule of sum is an intuitive principle stating that if there are a possible outcomes for an event (or ways to do something) and b possible outcomes for another event (or ways to do another thing), and the two events cannot both occur (or the two things can't both be done), then there are a + b total possible outcomes for the events (or total possible ways to do one of the things).
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 equality ((+)) = (()) can also be understood as an equivalence of different counting problems: the number of k-tuples of non-negative integers whose sum is n equals the number of (n + 1)-tuples of non-negative integers whose sum is k − 1, which follows by interchanging the roles of bars and stars in the diagrams representing configurations.
Where double counting involves counting one set in two ways, bijective proofs involve counting two sets in one way, by showing that their elements correspond one-for-one. The inclusion–exclusion principle , a formula for the size of a union of sets that may, together with another formula for the same union, be used as part of a double ...
Venn diagram showing the union of sets A and B as everything not in white. In combinatorics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
Number blocks, which can be used for counting. Counting is the process of determining the number of elements of a finite set of objects; that is, determining the size of a set. . The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the ...
Let n be very large and consider a random graph G on n vertices, where every edge in G exists with probability p = n 1/g −1. We show that with positive probability, G satisfies the following two properties: Property 1. G contains at most n/2 cycles of length less than g. Proof. Let X be the number cycles of length less than g.
In the celebrated application of counting trees (see below) and acyclic molecules, an arrangement of "colored beads" is actually an arrangement of arrangements, such as branches of a rooted tree. Thus the generating function f for the colors is derived from the generating function F for arrangements, and the Pólya enumeration theorem becomes a ...