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 PyPy3 (7.3.0) (judge verdict: Accepted)

# でつoO(YOU PLAY WITH THE CARDS YOU'RE DEALT..)
import sys
def main(N, A, X):
    # dp[i][j][k] = i番目のカードまで見て、k枚使って合計がjになる組み合わせの数
    dp = [[[0] * (N + 1) for __ in range(A * N + 5)] for _ in range(N + 1)]
    dp[0][0][0] = 1
    for i in range(1, N + 1):
        x = X[i - 1]
        for j in range(A * N + 1):
            for k in range(i + 1):
                dp[i][j][k] += dp[i - 1][j][k]
                if j >= x and k >= 1:
                    dp[i][j][k] += dp[i - 1][j - x][k - 1]

    ans = 0
    for n in range(1, N + 1):
        ans += dp[N][A * n][n]
    print(ans)

if __name__ == '__main__':
    input = sys.stdin.readline
    N, A = map(int, input().split())
    *X, = map(int, input().split())
    main(N, A, X)