# 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`

. more generally, computes the *gamma function* of `1+y`.

! 3 6 ! i.6 1 1 2 6 24 120

`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`. ,,y,,C,,x,, (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

### Common uses

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

### Related Primitives

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

### See Also

http://www.johndcook.com/blog/binomial_coefficients/ which discusses the domain of this operation. 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 ( ) x