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In combinatorics, double counting, also called two-way counting, is a proof technique that involves counting the size of a set in two ways in order to show that the two resulting expressions for the size of the set are equal. We describe a finite set X from two perspectives leading to two distinct expressions. Through the two perspectives, we demonstrate that each is to equal |X|. Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
The process necessarily provides a bijective mapping from the set to itself. This free bijection may very well be non-trivial; in certain theorems, the bijective mapping is more relevant than the expressions' equivalence. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...
Examples
Forming committees For instance, consider the number of ways in which a committee can be formed from a total of n people, with from 0 through to n members:
Method 1: There are two possibilities for each person - they may or may not be on the committee. Therefore there are a total of 2 × 2 × ... × 2 (n times) = 2n possibilities.
Method 2: The size of the committee must be some number between 0 and n. The number of ways in which a committee of k people can be formed from a total of n people is the binomial coefficient In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ...
Therefore the total number of ways is the sum of binomial coefficients over k = 0, 1, 2, ... n.
Equating the two expressions gives
Handshaking lemma An example of a theorem that is commonly proved with a double counting argument is the theorem that every graph contains an even number of vertices of odd degree. Let d(v) be the degree of vertex v. Every edge of the graph is incident to exactly two vertices, so by counting the number of edges incident to each vertex, we have counted each edge exactly twice. Therefore Graph theory is a growth area in mathematical research, and has a large specialized vocabulary. ...
where e is the number of edges. The sum of the degrees of the vertices is therefore an even number, which could not happen if an odd number of the vertices had odd degree. In mathematics, any integer (whole number) is either even or odd. ...
Sum of consecutive integers Suppose we have an (n + 1)×(n + 1) square of points. The number of points on the diagonal is exactly n + 1, and clearly the number of points S that are strictly above the diagonal equals the number of points strictly below the diagonal, so the total number of points in the square is n + 1 + 2S. On the other hand, the total number of points in the square is (n + 1)2, so - (n + 1)2 = n + 1 + 2S,
thus - n(n + 1) = 2S,
so - S = .
Further examples In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems; which often have a recursive flavour. ...
In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems; which often have a recursive flavour. ...
In combinatorial mathematics, Sperners lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors. ...
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