! Binomial

Also on this glyph

• r←!y — Factorial

r←x!y

r is the result of the scalar application of the binomial function, $\binom y x = \frac {y!} {x!(y - x)!}$, it represents the ways of picking x things from y things.

For non-positive integer arguments, the following operation is performed:

• if y is a negative integer:
  • if x is a positive integer, r is (¯1*x)×x!x-y+1
  • if x is an integer less than or equal to y, r is (¯1*y-x)×(y-x)!-k+1
  • if x is an integer, r is 0
  • if x is not an integer, fails
• if x and y are integers and if x is negative or greater than y, r is 0
• otherwise, r is (!y)÷x×⍥!y-x