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: Accepted)

#include <bits/stdc++.h>
 
#define int long long
 
#define Vanya Unstoppable
 
using namespace std;

const int N = 2507;

map<int, int> dp[57];

signed main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr); cout.tie(nullptr);
    
    int n, k;
    cin >> n >> k;
    
    int a[n + 1];
    for (int i = 1; i <= n; ++i) {
        cin >> a[i];
        a[i] = a[i] - k;
    }
    
    for (int i = 1; i <= n; ++i) {
        dp[i][a[i]] += 1;
        for (int j = -N; j <= N; ++j) {
            if (i + 1 > n) continue;
            int val = j + a[i + 1];
            if (val >= -N && val <= N) {
                dp[i + 1][val] += dp[i][j];
            }
            dp[i + 1][j] += dp[i][j];
        }
    }
    
    cout << dp[n][0];
}