English

Noun

combinations

1. Plural of combination
2. A type of cotton underwear for men, being a combination of a vest and trousers

In combinatorial mathematics, a combination is an un-ordered collection of unique sizes. (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, thus the set is the same as . 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 (also known as the "choose function"):

C^k_n = = \frac.

where n is the number of objects from which you can choose and k is the number to be chosen, and n! denotes the factorial.

As an example, the number of five-card hands possible from a standard fifty-two card deck is:

= \frac = \frac = \frac = 2598960.

The number of combinations with repetition can be calculated as:

= =

For example, if you have ten types of donuts (n) on a menu to choose from and you want three donuts (k) there are (10 + 3 − 1)! / 3!(10 − 1)! = 220 ways to choose (see also multiset).

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

= \frac \times \frac \times \frac \times \frac \times \cdots \times \frac .

Example:

= \frac \times \frac \times \frac \times \frac \times \frac = 2598960.

You get the same result for n-k as for k. Therefore, when k  is more than half of n, it may be easier to compute using n-k in place of k.

See also

• Factorial
• Combinadic (how to enumerate combinations and generate i:th combination in a reasonably way)
• Combinatorics
• Multiset
• Permutation
  • List of permutation topics
• Probability

External links

• Article on Combinations-PlainMath.Net Example and how to solve a combination
• Many Common types of permutation and combination math problems, with detailed solutions
• The Unknown Formula For combinations when choices can be repeated and order does NOT matter
• Web-based calculator of permutations and combinations