Difference between revisions of "Combination"
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<math>\sum_{x=0}^{n} \binom{n}{x} = 2^n</math> | <math>\sum_{x=0}^{n} \binom{n}{x} = 2^n</math> | ||
+ | One of the many proofs is by first inserting <math>a = b = 1</math> in to the [[binomial theorem]]. Because the combinations are the coefficients of <math>2^n</math>, and a and b cancel out because they are 1, the sum is <math>2^n</math>. | ||
== Examples == | == Examples == |
Revision as of 19:17, 3 November 2007
This is an AoPSWiki Word of the Week for Nov 1-7 |
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 .
Other Formulas
One of the many proofs is by first inserting in to the binomial theorem. Because the combinations are the coefficients of , and a and b cancel out because they are 1, the sum is .