Difference between revisions of "Combination"
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| − | == | + | == Introduction == |
| + | A '''combination''' is a way of choosing <math>r</math> objects from a set of <math>n</math> where the order in which the objects are chosen is irrelevant. We are generally concerned with finding the number of combinations of size <math>r</math> from an original set of size <math>n</math> | ||
| − | + | == Notation == | |
| − | |||
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The common forms of denoting the number of combinations of <math>{r}</math> objects from a set of <math>{n}</math> objects is: | The common forms of denoting the number of combinations of <math>{r}</math> objects from a set of <math>{n}</math> objects is: | ||
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* <math>{C}(n,r)</math> | * <math>{C}(n,r)</math> | ||
* <math>\,_{n} C_{r}</math> | * <math>\,_{n} C_{r}</math> | ||
| + | * <math> C^n_{r} </math> | ||
| − | + | == Formula == | |
<math>{{n}\choose {r}} = \frac {n!} {r!(n-r)!}</math> | <math>{{n}\choose {r}} = \frac {n!} {r!(n-r)!}</math> | ||
| − | + | == Derivation == | |
Consider the set of letters A, B, and C. There are <math>3!</math> 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 <math>{r}</math> objects from <math>{n}</math> elements <math>P(n,r)</math>, there are <math>{r}!</math> more ways to permute them than to choose them. We have <math>{r}!{C}({n},{r})=P(n,r)</math>, or <math>{{n}\choose {r}} = \frac {n!} {r!(n-r)!}</math>. | Consider the set of letters A, B, and C. There are <math>3!</math> 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 <math>{r}</math> objects from <math>{n}</math> elements <math>P(n,r)</math>, there are <math>{r}!</math> more ways to permute them than to choose them. We have <math>{r}!{C}({n},{r})=P(n,r)</math>, or <math>{{n}\choose {r}} = \frac {n!} {r!(n-r)!}</math>. | ||
| − | + | == Examples == | |
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45&year=2005&p=368223 AIME 2005II/1] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45&year=2005&p=368223 AIME 2005II/1] | ||
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45&year=2000&p=385889 AIME 2000II/5] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45&year=2000&p=385889 AIME 2000II/5] | ||
| − | + | == See also == | |
* [[Combinatorics]] | * [[Combinatorics]] | ||
* [[Permutations]] | * [[Permutations]] | ||
| + | * [[Pascal's Triangle]] | ||
Revision as of 22:50, 20 June 2006
Introduction
A combination is a way of choosing 
objects from a set of 
where the order in which the objects are chosen is irrelevant. We are generally concerned with finding the number of combinations of size from an original set of size
Notation
The common forms of denoting the number of combinations of 
objects from a set of 
objects is:
Formula
Derivation
Consider the set of letters A, B, and C. There are 
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 objects from elements 
, there are 
more ways to permute them than to choose them. We have 
, or .
