Yan Cui
I help clients go faster for less using serverless technologies.
This article is brought to you by
Don’t reinvent the patterns. Catalyst gives you consistent APIs for messaging, data, and workflow with key microservice patterns like circuit-breakers and retries for free.
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.
Whenever you’re ready, here are 3 ways I can help you:
- Production-Ready Serverless: Join 20+ AWS Heroes & Community Builders and 1000+ other students in levelling up your serverless game. This is your one-stop shop for quickly levelling up your serverless skills.
- I help clients launch product ideas, improve their development processes and upskill their teams. If you’d like to work together, then let’s get in touch.
- Join my community on Discord, ask questions, and join the discussion on all things AWS and Serverless.
let p15 =
let factorial n = [1I..n] |> List.fold (*) 1I
factorial(2I*20I)/(factorial(20I) ** 2)