Binomial
!

Binomial (!
) is a dyadic scalar function which gives the binomial coefficient between the two arguments. The argument order k!n
is reversed compared to most of traditional mathematical notation's alternative notations, for example and , but not others, like . Binomial shares the glyph !
with the monadic arithmetic function Factorial.
Examples
For nonnegative 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 kth value (zeroindexed) on the nth row (also zeroindexed) 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
Properties
The value of X!Y
equals (!Y)÷(!X)×!YX
.
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 , Binomial is defined to use the above equality for nonintegers. In that case, the Beta function becomes closely related to the Binomial, giving the identity ÷Y×(X1)!X+Y1
.
1 1.2 1.4 1.6 1.8 2!5
5 6.105689248 7.219424686 8.281104786 9.227916704 10
2!3j2
1J5