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)

import Control.Monad
import Data.Array.IO
import Data.Array.Unboxed
import qualified Data.Vector.Unboxed as V

type Idx = (Int,Int,Int)
type IODp = IOUArray Idx Int

main = do
    [n,a] <- map read.words<$>getLine::IO[Int]
    xs <- map read.words<$>getLine::IO[Int]
    let nx = n * maximum (a:xs)
    let vxs = V.fromList xs
    dp <- newArray ((0,0,0),(n,n,nx)) 0 :: IO IODp
    forM_ [(j,k,s)|s<-[0..nx],k<-[0..n],j<-[0..n]] $ \(j,k,s) -> do
        case (j,k,s) of
            (j,0,0) -> writeArray dp (j,0,0) 1
            (j,k,s) | j>0, s<vxs V.! (j-1) -> writeArray dp (j,k,s) =<< readArray dp (j-1,k,s)
            (j,k,s) | j>0, k>0, s>=vxs V.! (j-1) -> liftM2 (+) (readArray dp (j-1,k,s)) (readArray dp (j-1,k-1,s-vxs V.!(j-1))) >>= writeArray dp (j,k,s)
            otherwise -> return ()
    print.sum =<< mapM (\k -> readArray dp (n,k,k*a)) [1..n]

{-
dp(j,k,s) -> [x1..xj]からk集めた合計がsになる総数

dp(*,0,0) = 1
dp(j,k,s) = dp(j-1,k,s)                    , j>=1, s<xj
dp(j,k,s) = dp(j-1,k,s) + dp(j-1,k-1,s-xj) , k>=1, j>=1, s>=xj
other = 0
-}