Problem:
Score : 300 points

Problem Statement

Tak has N cards. On the i-th (1 \leq i \leq N) card is written an integer x_i.
He is selecting one or more cards from these N cards, so that the average of the integers written on the selected cards is exactly A.
In how many ways can he make his selection?

Constraints

1 \leq N \leq 50

1 \leq A \leq 50

1 \leq x_i \leq 50

N,\,A,\,x_i are integers.

Partial Score

200 points will be awarded for passing the test set satisfying 1 \leq N \leq 16.

Input

The input is given from Standard Input in the following format:

N A
x_1 x_2 ... x_N

Output

Print the number of ways to select cards such that the average of the written integers is exactly A.

Sample Input 1

4 8
7 9 8 9

Sample Output 1

5

The following are the 5 ways to select cards such that the average is 8:

Select the 3-rd card.

Select the 1-st and 2-nd cards.

Select the 1-st and 4-th cards.

Select the 1-st, 2-nd and 3-rd cards.

Select the 1-st, 3-rd and 4-th cards.

Sample Input 2

3 8
6 6 9

Sample Output 2

0

Sample Input 3

8 5
3 6 2 8 7 6 5 9

Sample Output 3

19

Sample Input 4

33 3
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Sample Output 4

8589934591

The answer may not fit into a 32-bit integer.

Solution in Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

import Control.Applicative
import Control.Monad
import Data.List
import Data.Array

solve :: Int -> Int -> [Int] -> Int
solve n a xs = sum $ [memo ! (n, k, k * a) | k <- [1 .. n]]
  where
    x = maximum (a:xs)
    xa = listArray (1, n) xs
    iter 0 0 0 = 1
    iter j k s
        | j >= 1, s < (xa ! j) =
            memo ! (j - 1, k, s)
        | j >= 1, k >= 1, s >= (xa ! j) =
            memo ! (j - 1, k, s) + memo ! (j - 1, k - 1, s - (xa ! j))
        | otherwise = 0
    memo = listArray
                ((0, 0, 0), (n, n, n * x))
                [ iter j k s
                | j <- [0 .. n]
                , k <- [0 .. n]
                , s <- [0 .. n * x]
                ]

main :: IO ()
main = do
    [n, a] <- fmap read . words <$> getLine :: IO [Int]
    xs <- fmap read . words <$> getLine :: IO [Int]
    print $ solve n a xs