99 questions/Solutions/26

(**) Generate the combinations of K distinct objects chosen from the N elements of a list

In how many ways can a committee of 3 be chosen from a group of 12 people? We all know that there are C(12,3) = 220 possibilities (C(N,K) denotes the well-known binomial coefficients). For pure mathematicians, this result may be great. But we want to really generate all the possibilities in a list.

-- Import the 'tails' function
--   > tails [0,1,2,3]
--   [[0,1,2,3],[1,2,3],[2,3],[3],[]]
import Data.List (tails)

-- The implementation first checks if there's no more elements to select,
-- if so, there's only one possible combination, the empty one,
-- otherwise we need to select 'n' elements. Since we don't want to
-- select an element twice, and we want to select elements in order, to
-- avoid combinations which only differ in ordering, we skip some
-- unspecified initial elements with 'tails', and select the next element,
-- also recursively selecting the next 'n-1' element from the rest of the
-- tail, finally consing them together

-- Using list comprehensions
combinations :: Int -> [a] -> [[a]]
combinations 0 _  = [ [] ]
combinations n xs = [ y:ys | y:xs' <- tails xs
                           , ys <- combinations (n-1) xs']

-- Alternate syntax, using 'do'-notation 
combinations :: Int -> [a] -> [[a]]
combinations 0 _  = return []
combinations n xs = do y:xs' <- tails xs
                       ys <- combinations (n-1) xs'
                       return (y:ys)

Another solution that's slightly longer, but doesn't depend on tails:

combinations :: Int -> [a] -> [[a]]
-- We don't actually need this base case; it's just here to
-- avoid the warning about non-exhaustive pattern matches
combinations _ [] = [[]]
-- Base case--choosing 0 elements from any list gives an empty list
combinations 0 _  = [[]]
-- Get all combinations that start with x, recursively choosing (k-1) from the
-- remaining xs. After exhausting all the possibilities starting with x, if there
-- are at least k elements in the remaining xs, recursively get combinations of k
-- from the remaining xs.
combinations k (x:xs) = x_start ++ others
    where x_start = [ x : rest | rest <- combinations (k-1) xs ]
          others  = if k <= length xs then combinations k xs else []

Variant: This version avoids getting a result consisting of the empty choice if we ask for, say, 10 elements from the empty list. (Instead, an empty list, suggesting "no options available" is returned.)

combinations :: Int -> [a] -> [[a]]
combinations 0 _ = [[]]
combinations _ [] = []
combinations n (x:xs) = (map (x:) (combinations (n-1) xs)) ++ (combinations n xs)