Difference between revisions of "Complementary counting"
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| − | '''Complementary counting''' is a method of counting where one counts what they don't want, then subtracts that from the total number of possibilities. In problems that involve complex or overly complicated and tedious [[casework]], complementary counting is often a far easier and more efficient approach. Within a problem statement, a large hint that complementary counting may lead to a quick solution is the phrase "not" or "at least". | + | '''Complementary counting''' is a method of [[counting]] where one counts what they don't want, then subtracts that from the total number of possibilities. In problems that involve complex or overly complicated and tedious [[casework]], complementary counting is often a far easier and more efficient approach. Within a problem statement, a large hint that complementary counting may lead to a quick solution is the phrase "not" or "at least". |
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. | ||
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* [[2004 AIME I Problems/Problem 15]] | * [[2004 AIME I Problems/Problem 15]] | ||
| − | + | == See also == | |
* [[Combinatorics]] | * [[Combinatorics]] | ||
* [[Probability]] | * [[Probability]] | ||
Revision as of 14:06, 17 May 2021
Complementary counting is a method of counting where one counts what they don't want, then subtracts that from the total number of possibilities. In problems that involve complex or overly complicated and tedious casework, complementary counting is often a far easier and more efficient approach. Within a problem statement, a large hint that complementary counting may lead to a quick solution is the phrase "not" or "at least".
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
