Beautiful combinatorial search

untitled.hs

import Control.Monad
import Data.Function

data Combination = Combination Int Int Int Int
    deriving Show

f (Combination x1 x2 x3 x4) = x1 + x2 + x3 - x4 >= 5

comb :: Maybe [Combination]
comb = sequence $ toComb <$> replicateM 4 [1..6] where
    toComb [x1,x2,x3,x4] = Just $ Combination x1 x2 x3 x4
    toComb _ = Nothing

c :: Integer -> Integer -> Integer
c n k = product [n-k+1..n] `div` product [1..k]

data Straightforward = Straightforward Int deriving Show
data Analytical = Analytical Integer deriving Show

main = do
    print $ Straightforward . length . filter f <$> comb
    print $ Analytical (c 16 4 - c 4 1 * c 10 4 + c 4 2 * c 4 4)