"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."
Introduces the fundamental methods of counting and combinatorial analysis such as Permutations and Combinations, the Multiplication Rule, and the Addition Rule.
Permutations: Calculating the number of arrangements of objects in a certain order.
Combinations: Calculating the number of ways to choose a certain number of objects regardless of order.
Factorials: Understanding the concept and computation of factorials.
Binomial coefficients: Calculating the number of ways to choose a certain number of objects from a larger set.
Venn diagrams: Understanding and using Venn diagrams to solve counting problems.
Inclusion-exclusion principle: Calculating the number of objects that belong to at least one of a set of categories.
Pigeonhole principle: Understanding and applying the concept of the pigeonhole principle in counting problems.
Generating functions: Understanding and utilizing generating functions to solve counting problems.
Recurrence relations: Understanding and utilizing recurrence relations to solve counting problems.
Probability: Understanding the basics of probability theory and its application in combinatorics.
Product Rule: The product rule states that if there are n ways to choose the first object and m ways to choose the second object, then there are n x m ways to choose both objects.
Sum Rule: The sum rule states that if there are n ways to choose one object and m ways to choose another object such that they cannot be chosen together, then there are n + m ways to choose an object.
Factorial Rule: The factorial rule states that if there are n distinct objects, then the number of ways to arrange them in a certain order is n!.
Permutation Rule: The permutation rule is used when selecting a certain number of objects from a larger set. In the case where r objects are being selected from a set of n objects, the number of permutations is nPr = n!/(n-r)!.
Combination Rule: The combination rule is used when selecting a certain number of objects from a larger set, and the order of selection is not important. In the case where r objects are being selected from a set of n objects, the number of combinations is nCr = n!/[(n-r)!r!].
Circular Permutation Rule: The circular permutation rule is used when objects are arranged in a circle. In this case, the number of permutations is (n-1)!.
Binomial Theorem: The binomial theorem relates to the expansion of a binomial (a + b) raised to a certain power. The expansion can be calculated using the formula (a + b)^n = ∑(k=0)^n nCk*a^(n-k)*b^k.
Inclusion-Exclusion Principle: The inclusion-exclusion principle is used to count the number of elements in a set that satisfy any one of several conditions. It states that the number of elements in the union of two or more sets can be calculated by adding the number of elements in each set, and then subtracting the number of elements in the intersection of each set.
Multinomial Theorem: The multinomial theorem relates to the expansion of a multinomial (a1 + a2 + ... + ak)^n. The expansion can be calculated using the formula ∑(k1+k2+...+km=n) nCk1,k2,...,km*a1^k1*a2^k2*...*ak^km.
Generating Functions: Generating functions are used to count the number of certain types of objects or events by converting a counting problem into a problem that involves manipulating a function. The coefficient of the term in the function can then be used to count the number of objects or events.
"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."