# Vocabulary/bang

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`! y`Factorial

Rank 0 *-- operates on individual atoms of y, producing a result of the same shape --*
WHY IS THIS IMPORTANT?

The *Factorial* of `y`.

Might be regarded as a special case of the *Stope* function (see link below), for start value identical to number of steps and step size `_1`: `n ^!._1 n` .

*/ 5 4 3 2 1 NB. product (5th 'falling power' of 5) 120 ! 5 NB. gets abbreviated 120 ! i.6 NB. 0! = 1 1 1 2 6 24 120

More generally, it computes the *Gamma function* of `(1+y)` (sometimes called the Pi function). If we defined `Ga(y)` as *Gamma* of real argument `(y)` we get

gamma=: ga=: !@<: : [: NB. monadic Ga(y)= (y-1)! ga _1r4p1 NB. Ga(-π/4) _5.42531 ga 0 NB. Ga(0) _ ga -i.6 _ __ _ __ _ __ ga 1 = ! 0 NB. Ga(1)= 1 = 0! 1 ga 1p1 NB. Ga(π) 2.28804 ga 2p1 NB. Ga(2π) 195.936 !(<:2p1) NB. (2π-1)! 195.936 exp=: ^ : [: NB. Exponential function, monad NB. calculating the (positive) intersections of exp(y)and ga(y) by NB. solving exp(y)- ga(y)= 0 numerically, using Newton-Raphson NB. with seed values 1 and 10, resp (exp-ga) VN^:_ (1) NB. IP4=~ (0.52;1.69) 0.524922 ga (exp-ga) VN^:_ (1) 1.69033 (exp-ga) VN^:_ (10) NB. IP5=~ (7.46;1743) 7.4636 ga (exp-ga) VN^:_ (10) NB. Gamma overtaking the Exponential 1743.42

`x ! y`Out Of

Rank 0 0 *-- operates on individual atoms of x and y, producing a result of the same shape --*
WHY IS THIS IMPORTANT?

Returns `y`*-Combinations-*`x`: ** yCx** (read:

`x`out of

`y`or:

`y`pick

`x`).

The number of ways of picking `x` balls (unordered) from a bag of `y` balls:

10!10 NB. There is only 1 way of picking all 10 balls 1 1!10 NB. There are 10 ways of choosing 1 ball from 10 10 2!10 NB. 45 ways to pick 2 out of 10 45 5!10 252

This example was found in K.E. Iverson, Computers and Mathematical Notation, E. Ambivalence; the table shows the first five rows of Pascal's Triangle:

0 1 2 3 4 !/ 0 1 2 3 4 NB. 'Out Of' table "with meaningful zeros" 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 |:!/~i.5 NB. (transposed view) 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1

### Common uses

In Probability theory, also Functional Analysis (Infinite Series).

### Details

In general `x ! y` is defined as `(!y) % (!x) * (!(y-x))` .

### Related Primitives

Stope Function (`^!._1 y`) to count permutations.

### See Also

John D. Cook, Binomial Coefficients discusses the domain of this operation, giving three definitions (from basic to general). There, `x!y` would be represented using a pixelated image, with x on the bottom and y on the top, both surrounded by extra large parenthesis. Something like this:

`{y \choose x}`

or `\binom{y}{x}`

in 'Math' (subset of LaTeX) notation.