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 ((<$>))

cartesianPower :: [Int] -> Int -> [[Int]]
cartesianPower xs 0 = []
cartesianPower xs 1 = [ [i] | i <- xs ]
cartesianPower xs n = [ i : x | i <- xs, x <- cartesianPower xs (n - 1) ]

solve iss a xs = filter (\is -> sum (map (xs !!) (snd $ unzip $ filter (\a -> fst a == 1) $ zip is [0 ..])) == (a * length (snd $ unzip $ filter (\a -> fst a == 1) $ zip is [0 ..]))) iss

main = do
  [n, a] <- fmap read <$> words <$> getLine :: IO [Int]
  xs <- fmap read <$> words <$> getLine :: IO [Int]
  putStrLn (show $ pred $ length $ solve (cartesianPower [0, 1] n) a xs)