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           Data.Array (Array, listArray, range, (!))
import qualified Data.ByteString.Char8 as B
import Data.List
import Data.Char

main :: IO ()
main = do
    [n, a] <- readInts
    xs     <- listArray (1, n) <$> readInts
    print $ solve n a xs

readInts :: IO [Int]
readInts = unfoldr (B.readInt . B.dropWhile isSpace) <$> B.getLine

solve :: Int -> Int -> Array Int Int -> Int
solve n a xs = sum [ dp n k (k * a) | k <- range (1, n) ]
  where
    dp 0 0 0 = 1
    dp j k s | j >= 1 && s < xs ! j            = dps ! (j - 1, k, s)
             | j >= 1 && k >= 1 && s >= xs ! j = dps ! (j - 1, k, s) + dps ! (j - 1, k - 1, s - xs ! j)
             | otherwise                       = 0

    dps      = listArray dpBounds [ dp j k s | (j, k, s) <- range dpBounds ]

    dpBounds = ((0, 0, 0), (n, n, n * xm))

    xm       = a `max` (maximum xs)