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)

#!/usr/bin/env python3

import itertools

N, A = list(map(int, input().split()))
x_list = list(map(int, input().split()))

max_x = max(max(x_list), A)
dp = [[[0 for s in range(N * max_x + 1)] for k in range(N + 1)] for j in range(N + 1)]
dp[0][0][0] = 1
ka_list = []

for j in range(N + 1):  # 何番目の数か
    x = x_list[j - 1]
    for k in range(N + 1):  # k枚選ぶとき
        for s in range(N * max_x + 1):
            if j == 0 and k == 0 and s == 0:
                dp[0][0][0] = 1
            elif j >= 1 and s < x:
                # オーバーするので選べない
                # ひとつ前と同じになる
                dp[j][k][s] = dp[j - 1][k][s]
            elif j >= 1 and k >= 1 and s >= x:
                dp[j][k][s] = dp[j - 1][k][s] + dp[j - 1][k - 1][s - x]
            else:
                dp[j][k][s] = 0

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