Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Combinatorics"
m
 
(43 intermediate revisions by 19 users not shown)
Line 1: Line 1:
−
'''Combinatorics''' is the study of counting.
+
'''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]]. Combinatorics also has many applications outside of pure mathematics, notably in [[theoretical computer science]], [[statistics]], and various fields of science.
  
−
== Introductory combinatorics ==
+
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.
−
The two most basic and fundamental ideas are that of [[permutations]] and [[combinations]].
 
−
In essentials, the permutation is the number of ways to create a subset of a larger set if order matters (i.e. A, B, C is different from A, C, B).
 
−
Similarly, the combination is the number of ways to create a subset of a larger set if order does NOT matter (i.e. A, B, C is the same as A, C, B).
 
  
 +
==Study Guides to Combinatorics==
 +
* [[Combinatorics/Introduction]]
 +
*[[Combinatorics/Intermediate]]
 +
*[[Combinatorics/Olympiad]]
  
−
'''Factorials'''
 
−
<math>n!=n(n-1)(n-2)\cdots(2)(1)</math>
 
  
−
A '''permutation''' of a set of r objects is any rearrangement of the r objects ''with regard to order''.  To find how many ways we can do this, note that for the first of the r elements, we have n different objects we can choose from.  For the second element there are (n-1) objects we can choose, (n-2) for the third and so on.  In general, the number of ways to permute r objects from a set of n is given by
 
−
<math>P(n,r)=n(n-1)(n-2)\cdots(n-r+1)=\frac{n!}{(n-r)!}</math>  (by the [[Fundamental Counting Principle]]
 
  
−
The number of '''combinations''' of r objects from a set of n total is the number of ways the r objects can be arranged ''with regard to order.''
 
−
Consider the set of letters A, B, and C.  There are P(3,3)=6 different permutations of those letters.  Since order doesn't matter with combinations, there is only one combination of those three.  In general, since for every permutation of r objects from n elements P(n,r), there are r! more ways to permute them than to choose them.  We have r!C(n,r)=P(n,r)or <math>\binom{n}{r}=\frac{n!}{r!(n-r)!}</math>
 
  
−
== Intermediate combinatorics ==
+
[[Combinatorics Challenge Problems]]
−
An important result of counting techniques is the formulation of the [[Principle of Inclusion-Exclusion]] (PIE).
 

Latest revision as of 01:12, 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



Combinatorics Challenge Problems

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Combinatorics&oldid=134587"