Difference between revisions of "Combination"
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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>. | ||
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| + | === Examples === | ||
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| + | * [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45&year=2005&p=368223 AIME 2005II/1] | ||
=== See also === | === See also === | ||
Revision as of 10:22, 19 June 2006
Definition
The number of combinations of 
objects from a set of 
objects is the number of ways the objects can be arranged with regard to order.
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 .
