Difference between revisions of "Combinatorics"
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'''Combinatorics''' is the study of [[discrete]] structures broadly speaking. Most notably, combinatorics involves studying the enumeration (counting) of said structures. For example, the number of three-[[cycle|cycles]] in a given [[graph]] is a combinatorial problem, as is the derivation of a non-[[recursive]] formula for the [[Fibonacci numbers]], and so too methods of solving the [[Rubiks cube]]. Mathematicians who spend their careers studying combinatorics are known as ''combinatorialists''. | '''Combinatorics''' is the study of [[discrete]] structures broadly speaking. Most notably, combinatorics involves studying the enumeration (counting) of said structures. For example, the number of three-[[cycle|cycles]] in a given [[graph]] is a combinatorial problem, as is the derivation of a non-[[recursive]] formula for the [[Fibonacci numbers]], and so too methods of solving the [[Rubiks cube]]. Mathematicians who spend their careers studying combinatorics are known as ''combinatorialists''. | ||
| − | Combinatorial problems often make up a good portion of problems found in mathematics competitions and can be approached by a variety of techniques, such as [[generating functions]] or the [[Principle of Inclusion-Exclusion|principle of inclusion-exclusion]]. | + | Combinatorial problems often make up a good portion of problems found in mathematics competitions and can be approached by a variety of techniques, such as [[generating functions]] or the [[Principle of Inclusion-Exclusion|principle of inclusion-exclusion]]. Combinatorics also has many applications outside of pure mathematics, notably in [[theoretical computer science]], [[statistics]], and various fields of science. |
People who encounter the term combinatorics for the first time often discredit it as "easy" because they "already know how to count." While this is true in the sense that people know how to count lists of numbers, enumeration problems are (typically) not nearly as simple as counting a list of numbers. One must first determine ''what'' and ''how'' something must be counted, both of which are often difficult to do. | People who encounter the term combinatorics for the first time often discredit it as "easy" because they "already know how to count." While this is true in the sense that people know how to count lists of numbers, enumeration problems are (typically) not nearly as simple as counting a list of numbers. One must first determine ''what'' and ''how'' something must be counted, both of which are often difficult to do. | ||
Latest revision as of 02:12, 4 October 2020
Combinatorics is the study of discrete structures broadly speaking. Most notably, combinatorics involves studying the enumeration (counting) of said structures. For example, the number of three-cycles in a given graph is a combinatorial problem, as is the derivation of a non-recursive formula for the Fibonacci numbers, and so too methods of solving the Rubiks cube. Mathematicians who spend their careers studying combinatorics are known as combinatorialists.
Combinatorial problems often make up a good portion of problems found in mathematics competitions and can be approached by a variety of techniques, such as generating functions or the principle of inclusion-exclusion. Combinatorics also has many applications outside of pure mathematics, notably in theoretical computer science, statistics, and various fields of science.
People who encounter the term combinatorics for the first time often discredit it as "easy" because they "already know how to count." While this is true in the sense that people know how to count lists of numbers, enumeration problems are (typically) not nearly as simple as counting a list of numbers. One must first determine what and how something must be counted, both of which are often difficult to do.
Study Guides to Combinatorics
