Binomial

The binomial coefficient , the number of k-element subsets of an n-element set.

WikipediaBinomial coefficientMathWorldBinomialCoefficientWikidataQ209875Encyclopedia of MathematicsBinomial_coefficientsBritannicatopic/binomial-coefficientDLMF26.31.2.iFungrimBinomial✓Fungrim entryOEISA073354Wolfram LanguageBinomial✓

Binomial(n, k)the binomial coefficient .

SageMathbinomial(n, k)SymPybinomial(n, k)

Domain: Combinatorics

Details

The general definition runs through the Gamma function: .
Symmetric in its arguments: .
Pascal's rule builds each row from the last: .
Row sums give .
compute-engine evaluates only integer n and k: k outside gives 0 for nonnegative n, while a negative n switches to the generalized falling-factorial formula rather than the Gamma form.

Examples

See also: Factorial, Multinomial, Pochhammer