"Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures."
The study of counting, permutations, combinations, and other discrete mathematical concepts used in solving problems.
Basic Counting Principles: Introduces the fundamental methods of counting and combinatorial analysis such as Permutations and Combinations, the Multiplication Rule, and the Addition Rule.
Binomial Theorem: A formula for expanding the powers of a binomial. It also shows how to compute coefficients of terms in the expansion.
Pascal's Triangle: A triangular array of binomial coefficients, used to solve problems in probability, algebra, and combinatorial analysis.
Inclusion-Exclusion Principle: A method for computing the size of a set formed by a union of other sets, by subtracting the size of their intersections to avoid double-counting.
Generating Functions: A formal power series used to encode sequences of numbers by representing them as coefficients of the series. Used in combinatorial analysis to study discrete structures.
Recurrence Relations: A sequence or function defined in terms of its previous term(s). Used to model dynamic processes and solve problems in combinatorics, graph theory, and computer science.
Catalan Numbers: A sequence of natural numbers that count various combinatorial objects such as balanced parentheses, binary trees, and noncrossing partitions.
Polya Enumeration Theorem: A method for counting certain types of combinatorial structures under symmetry by using permutation groups and generating functions.
"It has many applications ranging from logic to statistical physics and from evolutionary biology to computer science."
"Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry."
"In the later twentieth century, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right."
"One of the oldest and most accessible parts of combinatorics is graph theory."
"Graph theory... has numerous natural connections to other areas."
"Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms."
"Combinatorics is an area of mathematics primarily concerned with counting..."
"Combinatorics is primarily concerned with... certain properties of finite structures."
"Combinatorics is well known for the breadth of the problems it tackles."
"It is closely related to many other areas of mathematics..."
"[Combinatorics is] both as a means and an end in obtaining results..."
"Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context."
"Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context."
"Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms."
"Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry."
"In the later twentieth century, however, powerful and general theoretical methods were developed..."
"Many combinatorial questions have historically been considered in isolation..."
"...powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right."
"Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms."