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