! — Factorial / Binomial

Function. Hanzi: 阶/拣 (階/揀). Ranks: 0 monadic, 0 0 dyadic

!Y: Y!. Pervasive.

!5                ⍝ 120
!5ₓ               ⍝ 120ₓ

X!Y: binomial coefficient, Y choose X. Pervasive.

2!5               ⍝ 10
2ₓ!5ₓ             ⍝ 10ₓ

Extends through Gamma to complex arguments and to real arguments that aren’t whole. At a pole, a negative whole float, the result is an infinity or NaN, as the gamma function gives it. A negative machine integer gives the largest integer: i64’s under dynamic kinds, where integers widen, and the width’s own for fixed widths and under static kinds.

Past 100000, ! of a machine integer gives 0ₓ, and of a float ∞: !100001ₓ is 0ₓ, and !100001 is ∞. A ᵣ argument past 100000 raises LIMIT.