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 11:22, 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
.