Discover

In combinatorial mathematics, a 'combination' is an un-ordered collection of unique elements. (An ordered collection is called a permutation.) Given ''S'', the set of all possible unique elements, a combination is a subset of the elements of ''S''. The order of the elements in a combination is not important (two lists with the same elements in different orders are considered to be the same combination). Also, the elements cannot be repeated in a combination (every element appears uniquely once); this is often referred to as "without replacement/repetition". This is because combinations are defined by the elements contained in them, so the set {1, 1, 1} is the same as {1}. For example, from a 52-card deck any 5 cards can form a valid combination (a hand). The order of the cards doesn't matter and there can be no repetition of cards.
A ''k''-combination (or ''k''-subset) is a subset with ''k'' elements. The number of ''k''-combinations (each of size ''k'') from a set ''S'' with ''n'' elements (size ''n'') is the binomial coefficient
:
As an example, the number of five-card hands possible from a standard fifty-two card deck is:
:
A combination is a special case of a partition of a set; specifically, a partition into two sets of size ''k'' and ''n'' − ''k''.
Since it is impractical to calculate $n!$ if the value of ''n'' is very large, a more efficient algorithm is
:
Example:
:

Contents

See also

External links

See also

★ Combinadic

★ Combinatorics

★ Multiset

★ Permutation

★
★ List of permutation topics

★ Probability

External links

★ Excellent Review of Combinations-PlainMath.Net Example and how to solve a combination

★ Many Common types of permutation and combination math problems, with detailed solutions