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Encyclopedia > Combinations

In combinatorial mathematics, a combination of members of a set is a subset. A k-combination is a subset of S with k elements. The order of listing the elements is not important in combinations: two lists with the same elements in different orders are considered to be the same combination. The number of k-combinations or k-subsets of set with n elements is the binomial coefficient "n choose k", written as nCk, nCk or as

or occasionally as C(n, k).

One method of deriving a formula for nCk proceeds as follows:

  1. Count the number of ways in which one can make an ordered list of k different elements from the set of n. This is equivalent to calculating the number of k-permutations.
  2. Recognizing that we have listed every subset many times, we correct the calculation by dividing by the number of different lists containing the same k elements:

Since

(see factorial), we find

It is useful to note that C(n, k) can also be found using Pascal's triangle, as explained in the binomial coefficient article.


See also


  Results from FactBites:
 
Combination lock - Wikipedia, the free encyclopedia (618 words)
A combination lock is a type of lock in which a sequence of numbers or symbols is used to open the lock.
The notches on the disc correspond to the numerals in the correct combination.
For example, early combination padlocks made by Master lock could be cracked by pulling on the shackle of the lock and turning the dial until it stopped; each numeral in the combination could be revealed in this manner.
Combination - Wikipedia, the free encyclopedia (163 words)
It has been suggested that Permutations and combinations be merged into this article or section.
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).
  More results at FactBites »


 

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