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: Accepted)

{-# LANGUAGE RankNTypes #-}                                                                                                                           
import Control.Monad                                                                                                                                  
import Control.Monad.ST                                                                                                                               
import Data.Array.ST                                                                                                                                  
                                                                                                                                                      
type Idx = (Int,Int,Int)                                                                                                                              
                                                                                                                                                      
main :: IO ()                                                                                                                                         
main = do                                                                                                                                             
  [n,a] <- map read . words <$> getLine :: IO [Int]                                                                                                   
  xs <- (0:) . map read . words <$> getLine :: IO [Int]                                                                                               
  print $ sum $ solve n a xs                                                                                                                          
                                                                                                                                                      
solve :: Int -> Int -> [Int] -> [Int]                                                                                                                 
solve m b ys = runST $ do                                                                                                                             
  let smax = m * maximum (b:ys)                                                                                                                       
  dptbl <- newArray ((0,0,0),(m,m,smax)) (-1) :: forall s. ST s (STUArray s Idx Int)                                                                  
  forM_ [(jt,k,s) | s <- [0..smax], k <- [0..m], jt <- zip [0..] ys] $ \((j,yj),k,s) -> do                                                            
    ra <- readArray dptbl (j,k,s)                                                                                                                     
    case ra of                                                                                                                                        
      -1 | (j,k,s) == (0,0,0)    -> writeArray dptbl (j,k,s) 1                                                                                        
         | j>=1 && s<yj          -> readArray dptbl (j-1,k,s) >>= writeArray dptbl (j,k,s)                                                            
         | j>=1 && k>=1 && s>=yj -> liftM2 (+) (readArray dptbl (j-1,k,s)) (readArray dptbl (j-1,k-1,s-yj)) >>= writeArray dptbl (j,k,s)              
         | otherwise             -> writeArray dptbl (j,k,s) 0                                                                                        
      _                          -> return ()                                                                                                         
  mapM (\k -> readArray dptbl (m,k,k*b)) [1..m]