commented factorial:

::  produce a gate w/ sample n, an unsigned decimal number
::
|=  n=@ud
::  combine unsigned decimal 1 w/ face t with the subject
::
=/  t=@ud  1
::  produce and evaluate trap. Starting point for recursion.
::
|-
::  If n is equal to 1, return t, the completed factorial
::
?:  =(n 1)
    t
::  else tail recurse, decrement n, t as running product of the factorial
::
$(n (dec n), t (mul t n))

return third element of list without standard library list functions:

|=  input=(list @)
=/  count  2
|-
?:  =(count 0)
    ?~(input ~ i.input)
$(input ?~(input ~ t.input), count (dec count))