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 <bits/stdc++.h>
#define INF 5000000000000000000
#define ll long long
#define pll pair<ll, ll>
using namespace std;

ll N, A;
void dfs(ll now, ll sum, ll num_used, vector<ll>& x, map<ll, ll>& num) {
  if (now == x.size()) {
    num[A * num_used - sum] += 1;
  }
  else {
    dfs(now + 1, sum, num_used, x, num);
    dfs(now + 1, sum + x.at(now), num_used + 1, x, num);
  }
}

int main() {
  cin >> N >> A;
  vector<ll> x1(N / 2), x2(N - N / 2);
  for (ll i = 0; i < N / 2; ++i) {
    cin >> x1.at(i);
  }
  for (ll i = 0; i < (N - N / 2); ++i) {
    cin >> x2.at(i);
  }
  map<ll, ll> num1, num2;
  dfs(0, 0, 0, x1, num1);
  dfs(0, 0, 0, x2, num2);
  // for (ll i = 0; i < num1.size(); ++i) {
  //   cout << num1.at(i) << " ";
  // }
  // cout << "\n";
  // for (ll i = 0; i < num2.size(); ++i) {
  //   cout << num2.at(i) << " ";
  // }
  // cout << "\n";
  ll ans = 0;
  for (const auto& [key, value] : num1) {
    ans += value * num2[-key];
  }
  
  ans -= 1;
  cout << ans << "\n";
}