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: Time Limit Exceeded)


import numpy as np

def read_input():
    n, a = map(int, input().split())
    xlist = list(map(int, input().split()))
    return n, a, xlist


def submit():
    n, a, xlist = read_input()

    dp = np.zeros((n + 1, n + 1, (n + 1) * max(max(xlist), a)))

#   dp[j][k][s] : j枚目までのカードでk枚選び、合計sにする方法の数
#         (1) dp[0][0][0] : 1枚も選ばずに合計0にする方法は1種類
#               j = k = s = 0のとき
#         (2) dp[j][k][s] = dp[j - 1][k][s] if xj > s : j枚目のカードがsより大きいとき、sを作りようがないのでj-1枚目までの手法と一緒
#               j > 0, xj > sのとき
#         (3) dp[j][k][s] = dp[j - 1][k][s] + dp[j - 1][k - 1][s - xj]
#               : xj <= sのとき、j-1枚目まででsを作る方法と、j-1枚目まででs-xjをつくり、j枚目を選んでk枚にする
#               j > 0, k > 0, s >= xjのとき
#         (4) dp上記以外は無理なので0

    for j in range(n + 1):
        for k in range(n + 1):
            for s in range((n + 1) * max(xlist)):
                if j == k == s == 0:
                    dp[j][k][s] = 1
                elif j > 0 and xlist[j - 1] > s:
                    dp[j][k][s] = dp[j - 1][k][s]
                elif j > 0 and k > 0 and xlist[j - 1] <= s:
                    dp[j][k][s] = dp[j - 1][k][s] + dp[j - 1][k - 1][s - xlist[j - 1]]
                else:
                    dp[j][k][s] = 0

    methods = 0
    for i in range(1, n + 1):
        methods += dp[n][i][i * a]

    print(int(methods))


if __name__ == '__main__':
    submit()