Difference between revisions of "Complementary counting"
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| − | In [[combinatorics]], ''' | + | In [[combinatorics]], '''complementary counting''' is a [[counting]] method where one counts what they don't want, then subtracts that from the total number of possibilities. In problems that involve complex or tedious [[casework]], complementary counting is often a far simpler approach. A large hint that complementary counting may lead to a quick solution is the phrase "not" or "at least" within a problem statement. |
More formally, if <math>B</math> is a subset of <math>A</math>, complementary counting exploits the property that <math>|B| = |A| - |B'|</math>, where <math>B'</math> is the [[complement]] of <math>B</math>. In most instances, though, <math>A</math> is obvious from context. | More formally, if <math>B</math> is a subset of <math>A</math>, complementary counting exploits the property that <math>|B| = |A| - |B'|</math>, where <math>B'</math> is the [[complement]] of <math>B</math>. In most instances, though, <math>A</math> is obvious from context. | ||
Revision as of 15:37, 18 May 2021
In combinatorics, complementary counting is a counting method where one counts what they don't want, then subtracts that from the total number of possibilities. In problems that involve complex or tedious casework, complementary counting is often a far simpler approach. A large hint that complementary counting may lead to a quick solution is the phrase "not" or "at least" within a problem statement.
More formally, if 
is a subset of 
, complementary counting exploits the property that 
, where 
is the complement of . In most instances, though, is obvious from context.
Video
This is a video explaining the basics of casework, complementary counting, and overcounting (more specifically, the Principle of Inclusion-Exclusion): https://youtu.be/Zhsb5lv6jCI
