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: Wrong Answer)

#include <bits/stdc++.h>

using namespace std;
using ll = long long;

int solve(vector<int> &vx, int offset, int n, int sum,
          vector<vector<vector<ll>>> &memo) {
  if (offset == vx.size()) {
    return sum == 0 && n == 0;
  }

  if (n < 0 || sum < 0) {
    return 0;
  }

  if (memo[offset][n][sum] == -1) {
    memo[offset][n][sum] = solve(vx, offset + 1, n, sum, memo) +
                           solve(vx, offset + 1, n - 1, sum - vx[offset], memo);
  }

  return memo[offset][n][sum];
}

int main(int argc, const char *argv[]) {
  int n, a;
  cin >> n >> a;

  vector<int> vx(n);
  for (int i = 0; i < n; ++i) {
    cin >> vx[i];
  }

  vector<vector<vector<ll>>> memo(
      60, vector<vector<ll>>(60, vector<ll>(60 * 60, -1)));
  ll ans = 0;
  for (int i = 1; i <= n; ++i) {
    ans += solve(vx, 0, i, a * i, memo);
  }

  cout << ans << '\n';
}