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)

import Data.Map.Strict ((!))
import qualified Data.Map.Strict as A

main :: IO ()
main = do
    [n, a] <- map read . words  <$> getLine :: IO [Int]
    ns <- map read . words  <$> getLine :: IO [Int]
    print $ dp a n ns

dp :: Int -> Int -> [Int] -> Int
dp a n ns' = sum [memo!(j, k)|j<-[1..n], k<-[0..50*n], mod k j == 0, div k j == a] 
    where
    ns = A.fromList [(i,x)|(i,x) <- zip [1..] ns']
    memo = dpIter 0 A.empty
    dpIter :: Int -> A.Map (Int, Int) Int -> A.Map (Int, Int) Int
    dpIter i memo
        | i <  n = dpIter (i+1) $ A.fromList [((j,k), f i j k)|j<-[0..n], k<-[0..50*n]]
        | i == n =                A.fromList [((j,k), f i j k)|j<-[0..n], k<-[0..50*n]]
        where
        -- f (nsのインデックス) (使う枚数) (値) = 何組あるか
        f 0 0 0 = 1
        f 0 _ _ = 0
        f i j k 
            | j>0 && k-(ns!i) >= 0 = (memo!(j,k)) + (memo!(j-1,k - ns!i))
            | otherwise            = memo!(j,k)