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++ (GCC 9.2.1) (judge verdict: Time Limit Exceeded)

#include <iostream>
#include <vector>
#define MAX 100000000

using namespace std;


int n,a;
vector <int> c;
int X[50];
int sum_array[MAX];


void Try(int k,int l,int r) {
    if(k == r) {
        int sum = 0;
        for(int i=0;i<r;i++) sum += sum_array[i];
        if(sum/r == a && sum % r ==0) c.push_back(sum);
        return;
    }
    if(l>=n) return;
    sum_array[k] = X[l];
    Try(k+1,l+1,r);
    Try(k,l+1,r);
}

int main()
{
    cin >> n >> a;
    for(int i=0;i<n;i++) cin >> X[i];
    for(int i=1;i<=n;i++) Try(0,0,i);
    cout << c.size() << endl;
    return 0;
}