The solution, of course, would be too easy if I could just use 20!40 in APL. This number is too large for 32-bit integers, so it goes to floating point, and I lose two digits that I need. I could have done this the long way, but I decided that this would be a great chance to play with the Foreign Function interface to Java.
∇R←PEFIFTEEN;N;K;FACT;TIMES;NBI;STR;DIV;⎕IO ⎕IO←1 ⍝ Compute the number of paths through a 20×20 grid ⍝ which is just 20!40. N←40 ⋄ K←20 ⊣⎕FX 'R←NBI X' 'R←''java'' ⎕NEW ''java.math.BigInteger''(⍕X)' ⊣⎕FX 'R←X TIMES Y' 'R←X.multiply Y' ⊣⎕FX 'R←FACT N' 'R←TIMES/NBI¨⍳N' ⊣⎕FX 'R←STR J' 'R←J.toString' ⊣⎕FX 'R←X DIV Y' 'R←X.divide Y' R←STR(FACT N)DIV(FACT K)TIMES FACT N-K ∇