The number of ways of picking k unordered outcomes from n possibilities. Also known as the binomial coefficient or choice number and read "n choose k,"

 _nC_k=(n; k)=(n!)/(k!(n-k)!),

where n! is a factorial (Uspensky 1937, p. 18). For example, there are (4; 2)=6 combinations of two elements out of the set {1,2,3,4}, namely {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, and {3,4}. These combinations are known as k-subsets.

The number of combinations (n; k) can be computed in the Wolfram Language using Binomial[n, k], and the combinations themselves can be enumerated in the Wolfram Language using Subsets[Range[n],{k}].

Muir (1960, p. 7) uses the nonstandard notations (n)_k=(n; k) and (n^_)_k=(n-k; k).

See also

Ball Picking, Binomial Coefficient, Choose, Derangement, Factorial, k-Subset, Multichoose, Multinomial Coefficient, Multiset, Permutation, String, Subfactorial

Explore with Wolfram|Alpha


Conway, J. H. and Guy, R. K. "Choice Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 67-68, 1996.Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960.Ruskey, F. "Information on Combinations of a Set.", S. "Combinations." §1.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 40-46, 1990.Uspensky, J. V. Introduction to Mathematical Probability. New York: McGraw-Hill, p. 18, 1937.

Referenced on Wolfram|Alpha


Cite this as:

Weisstein, Eric W. "Combination." From MathWorld--A Wolfram Web Resource.

Subject classifications