Combinatorial principles

In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule of product, and inclusion–exclusion principle are often used for enumerative purposes. Bijective proofs are utilized to demonstrate that two sets have the same number of elements. The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a discrete context. Many combinatorial identities arise from double counting methods or the method of distinguished element. Generating functions and recurrence relations are powerful tools that can be used to manipulate sequences, and can describe if not resolve many combinatorial situations. Rule of sum 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). More formally, the sum of the sizes of two disjoint sets is equal to the size of their union. Rule of product The rule of product is another intuitive principle stating that if there are a ways to do something and b ways to do another thing, then there are a · b ways to do both things. Inclusion–exclusion principle The inclusion–exclusion principle relates the size of the union of multiple sets, the size of each set, and the size of each possible intersection of the sets. The smallest example is when there are two sets: the number of elements in the union of A and B is equal to the sum of the number of elements in A and B, minus the number of elements in their intersection. Generally, according to this principle, if A1, ..., An are finite sets, then Rule of division (combinatorics) The rule of division states that there are n/d ways to do a task if it can be done using a procedure that can be carried out in n ways, and for every way w, exactly d of the n ways correspond to way w.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (4)

Related courses (2)

Related lectures (51)

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a

MATH-101(e): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

Combinatorial Proofs: Frobenius Theorem MATH-494: Topics in arithmetic geometry

Explores combinatorial proofs for the Frobenius Theorem, demonstrating mathematical reasoning and calculations.

Algebraic Identities and Trigonometry MATH-101(e): Analysis I

Covers algebraic identities, trigonometry, and real functions, including injective, surjective, bijective, and reciprocal functions.

Orthogonal Projections and Reflections in 2D

Covers the geometric description of orthogonal projections and reflections in 2D, focusing on transformations and their properties.

Combinatorial principles

Enumerative combinatorics

Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in Sn for each n.

Combinatorial proof

In mathematics, the term combinatorial proof is often used to mean either of two types of mathematical proof: A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in two different ways to obtain the different expressions in the identity. Since those expressions count the same objects, they must be equal to each other and thus the identity is established. A bijective proof. Two sets are shown to have the same number of members by exhibiting a bijection, i.