Binomial

is a dyadic scalar function which gives the binomial coefficient $$\tbinom nk$$ between the two arguments. The argument order  is reversed compared to most of traditional mathematical notation's alternative notations, for example $$C(n,k)$$ and $$_nC_k$$, but not others, like $$C_n^k$$. Binomial shares the glyph  with the monadic arithmetic function Factorial.

Examples
For non-negative integer arguments, the binomial coefficient  is equal to the number of ways to choose k items out of n distinct items. For example,  is 10 because there are 10 ways to pick 3 items out of 5: 123, 124, 125, 134, 135, 145, 234, 235, 245, 345.

also corresponds to the k-th value (zero-indexed) on the n-th row (also zero-indexed) of Pascal's triangle.

Properties
The value of  equals.

In multiple implementations where Factorial is extended to use the Gamma function $$\Gamma(n)$$, Binomial is defined to use the above equality for non-integers. In that case, the Beta function $$\Beta(x,y)$$ becomes closely related to the Binomial, giving the identity $$\Beta(X,Y)$$.

Documentation

 * Dyalog
 * APLX
 * J Dictionary, NuVoc