#### Problem

Starting in the top left corner of a 2×2 grid, there are 6 routes (without backtracking) to the bottom right corner.

How many routes are there through a 20×20 grid?

#### Solution

let rec factorial(n:bigint) = if n <= 1I then 1I else n * factorial(n-1I) let combo n k = factorial(n) / (factorial(k) * factorial(n-k)) let answer = combo 40I 20I

It took me a while to figure out that this problem is actually a simple combination problem – consider a X by Y grid, any route from the top left to the bottom right corner without backtracking must have travelled Right X number of times and Down Y number of times. In the case of the original example:

RRDD, RDRD, RDDR, DRRD, DRDR, DDRR

This also means that all routes have a total of X + Y steps, and the number of routes is equal to the number of ways you can pick X number of R moves out of X + Y, i.e.

where n = X + Y and k = X.

let p15 =

let factorial n = [1I..n] |> List.fold (*) 1I

factorial(2I*20I)/(factorial(20I) ** 2)