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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>

using namespace std;

const int N = 55;

int n, a, x;
long long f[N][N * N][N];

int main() {
    cin >> n >> a;
    f[0][0][0] = 1;
    for (int i = 1; i <= n; ++i) {
        cin >> x;
        for (int sum = 0; sum <= (i - 1) * 50; ++sum) {
            for (int cnt = 0; cnt <= (i - 1); ++cnt) {
                f[i][sum][cnt] += f[i - 1][sum][cnt];
                f[i][sum + x][cnt + 1] += f[i - 1][sum][cnt];
            }
        }
    }
    long long result = 0;
    for (int sum = 0; sum <= n * 50; ++sum) {
        for (int cnt = 1; cnt <= n; ++cnt) {
            if (sum == cnt * a)
                result += f[n][sum][cnt];
        }
    }
    cout << result << endl;
    return 0;
}