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)

{-# LANGUAGE BangPatterns #-}
import Data.List
import Data.Maybe
import Data.Function (fix)
import qualified Data.ByteString as BS
import qualified Data.ByteString.Char8 as BS8
 
getInts :: IO [Int]
getInts = (map parseInt . BS8.words) <$> BS.getLine 
    where parseInt = fst . fromJust . BS8.readInt

data Tree a = Tree (Tree a) a (Tree a)

instance Functor Tree where
    fmap f (Tree l m r) = Tree (fmap f l) (f m) (fmap f r)

index :: Tree a -> Int -> a
index (Tree _ m _) 0 = m
index (Tree l _ r) n = case (n -1) `divMod` 2 of
    (q,0) -> index l q
    (q,1) -> index r q

nats :: Tree Int
nats = go 0 1
    where go !n !s = Tree (go l s') n (go r s')
            where l = n + s
                  r = l + s
                  s' = s * 2

memo :: (Int -> a) -> (Int -> a)
memo f = index cached
    where cached = fmap f nats

memo2 :: (Int -> Int -> a) -> (Int -> Int -> a)
memo2 f = memo (\x -> memo (f x))

memo3 :: (Int -> Int -> Int -> a) -> (Int -> Int -> Int -> a)
memo3 f = memo (\x -> memo2 (f x))

{- dp example
fibo 0 = 0
fibo 1 = 1
fibo n = mfibo (n-1) + mfibo (n-2)

mfibo = memo f
-}


solve :: [Int] -> Int -> Integer
solve arr a = sum $ map f [n,(n-1)..1]
    where n = length arr
          f cnt = f' 0 cnt (cnt*a)
          f' idx cnt s
            | idx == n && (cnt == 0 && s == 0) = 1
            | idx == n = 0
            | otherwise = f'' (idx+1) cnt s + f'' (idx+1) (cnt-1) (s-(arr !! idx))
          f'' idx cnt s 
            | cnt < 0 || s < 0 = 0
            | otherwise = (memo3 f') idx cnt s

main :: IO ()
main = do
    [n, a] <- getInts
    arr <- getInts
    print (solve arr a)