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

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

main = do
    [n,a] <- map read.words<$>getLine::IO[Int]
    ys <- map (flip(-)a.read).words<$>getLine::IO[Int]
    let nx = n * maximum (a:ys)
    dp <- newArray ((0,-nx),(n,nx)) 0 :: IO IODp
    mapM_ (f dp nx) [(j,t)|j<-zip [0..] (0:ys), t<-[-nx..nx]]
    print.flip(-)1 =<< readArray dp (n,0)

f dp _ ((0,_),0) = writeArray dp (0,0) 1
f dp nx ((j,yj),t) | j>0, -nx<=t-yj, t-yj<=nx = liftM2 (+) (readArray dp (j-1,t)) (readArray dp (j-1,t-yj)) >>= writeArray dp (j,t)
f dp _ ((j,yj),t) | j>0 = readArray dp (j-1,t) >>= writeArray dp (j,t)
f _ _ _ = return ()

{-
dp(j,t) -> [y1..yj]から合計をtにするような選び方の総数, -nx<=t<=nx

dp(0,0) = 1
dp(j,t) = dp(j-1,t) + dp(j-1,t-yj) , j>=1, -nx <= t-yj <= nx
dp(j,t) = dp(j-1,t)                , j>=1
other = 0
-}