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 Python (3.4.3) (judge verdict: Runtime Error)

from itertools import product
N, A = map(int, input().split())
x = list(map(int, input().split()))
x_max = max(x)

dp = [[[0 for s in range(x_max * N + 1)] for n in range(N + 1)] for i in range(N + 1)]
dp[0][0][0] = 1

for i, n, s in product(range(1, N + 1), range(N + 1), range(x_max * N + 1)):
    if s < x[i - 1]:
        dp[i][n][s] = dp[i - 1][n][s]
    elif 0 < n and x[i - 1] <= s:
        dp[i][n][s] = dp[i - 1][n][s] + dp[i - 1][n - 1][s - x[i - 1]]

ans = sum([dp[N][n][n*A] for n in range(1, N + 1)])
print(ans)