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)

    -- AtCoder My Practice
    -- author: Leonardone @ NEETSDKASU
     
    -- 解説読後
    -- http://arc060.contest.atcoder.jp/data/arc/060/editorial.pdf
     
    import qualified Data.Array as A
    import qualified Data.Map as M
     
    main = print . solve . map read . words =<< getContents
     
    solve (n:a:xs) = ans where
        x  = maximum (a:xs)
        nx = n * x
        ys = A.listArray (1, n) $ map (+ (-a)) xs
        dp j t m =
            case M.lookup (j, t) m of
            Just c  -> (c, m)
            Nothing ->
                case True of
                    _ | j == 0 && t == nx -> (1, m)
                    _ | j >= 1 && (ty <  0 || ty >  2 * nx) -> ans where
                        (a1, m1) = dp j1 t m
                        ans = (a1, M.insert (j, t) a1 m1) -- 間違いあったのに通った？
                    _ | j >= 1 && (ty >= 0 && ty <= 2 * nx) -> ans where
                        (a1, m1) = dp j1 t m
                        (a2, m2) = dp j1 ty m1
                        ans = (a1 + a2, M.insert (j, t) (a1 + a2) m2)
                    _ -> (0, m)
            where
                j1 = j - 1
                ty = t - (ys A.! j)
        ans = fst (dp n nx M.empty) - 1