Difference between revisions of "Combination"

Revision as of 12:21, 28 September 2007

A combination is a way of choosing $r$ objects from a set of $n$ where the order in which the objects are chosen is irrelevant. We are generally concerned with finding the number of combinations of size $r$ from an original set of size $n$

Notation

The common forms of denoting the number of combinations of ${r}$ objects from a set of ${n}$ objects is:

$\binom{n}{r}$
${C}(n,r)$
$\,_{n} C_{r}$
$C_n^{r}$

Formula

${{n}\choose {r}} = \frac {n!} {r!(n-r)!}$

Derivation

Consider the set of letters A, B, and C. There are $3!$ 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 ${r}$ objects from ${n}$ elements $P(n,r)$ , there are ${r}!$ more ways to permute them than to choose them. We have ${r}!{C}({n},{r})=P(n,r)$ , or ${{n}\choose {r}} = \frac {n!} {r!(n-r)!}$ .

Examples

Links

[1]

@@ Line 25: / Line 25: @@
 * [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45&year=2000&p=385889 AIME 2000II/5]
+==Links==
+* [http://arml.yanco.com/20012002/armlcombinatorics.pdf]
 == See also ==

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