# Binomial

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 !

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

## Examples

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

      0 1 2 3 4 5!5
1 5 10 10 5 1

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

      ⍉∘.!⍨ 0,⍳5
1 0  0  0 0 0
1 1  0  0 0 0
1 2  1  0 0 0
1 3  3  1 0 0
1 4  6  4 1 0
1 5 10 10 5 1
Works in: Dyalog APL

## Properties

The value of X!Y equals (!Y)÷(!X)×!Y-X.

      Alt←{(!⍵)÷(!⍺)×!⍵-⍺}
0 1 2 3 4 5 Alt 5
1 5 10 10 5 1

In multiple implementations where Factorial is extended to use the Gamma function ${\displaystyle \Gamma (n)}$, Binomial is defined to use the above equality for non-integers. In that case, the Beta function ${\displaystyle \mathrm {B} (x,y)}$ becomes closely related to the Binomial, giving the identity ${\displaystyle \mathrm {B} (X,Y)}$ ${\displaystyle \Leftrightarrow }$ ÷Y×(X-1)!X+Y-1.

      1 1.2 1.4 1.6 1.8 2!5
5 6.105689248 7.219424686 8.281104786 9.227916704 10
2!3j2
1J5
Works in: Dyalog APL