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)

-- AtCoder My Practice
-- author: Leonardone @ NEETSDKASU

-- 解説読後
-- http://arc060.contest.atcoder.jp/data/arc/060/editorial.pdf

-- 満点解法1
-- 解説にtypo dp[N][k][kA] が正しい

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)
    ys = A.listArray (1, n) xs
    dp j k s m =
        case M.lookup (j, k, s) m of
        Just c  -> (c, m)
        Nothing ->
            case True of
                _ | j == 0 && k == 0 && s == 0 -> (1, m)
                _ | j >= 1 && sx < 0 -> ans where
                    (a1, m1) = dp j1 k s m
                    ans = (a1, M.insert (j, k, s) a1 m1)
                _ | j >= 1 && k >= 1 && sx >= 0 -> ans where
                    (a1, m1) = dp j1 k s m
                    (a2, m2) = dp j1 (k - 1) sx m1
                    ans = (a1 + a2, M.insert (j, k, s) (a1 + a2) m2)
                _ -> (0, m)
            where
                j1 = j - 1
                sx = s - (ys A.! j)
    ans = fst $ foldl (\(c, m) k -> let (a1, m1) = dp n k (k * a) m in (c + a1, m1)) (0, M.empty) [1 .. n]