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.
⍝ 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