Combinations and Permutations

A combination is an unordered 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 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.

Example of permutations:
1,2,3
1,3,2
2,1,3
2,3,1
...
n! (3 factorial is the number of permutations)

Example of combinations:
Subsets chosen from a larger set of objects in which the order of the items in the subset does not matter. For example, determining how many different committees of four persons could be chosen from a set of nine persons