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;

int main() {
  int n, a;
  cin >> n >> a;
  vector<int> x(n);
  for (int i = 0; i < n; i++) {
    cin >> x.at(i);
    x.at(i) -= a;
  }
  vector<vector<int64_t>> dp(n + 1, vector<int64_t>(100 * n));
  dp.at(0).at(50 * n) = 1;
  for (int i = 0; i < n; i++) {
    for (int j = 0; j < 100 * n; j++) {
      dp.at(i + 1).at(j) += dp.at(i).at(j);
      int k = j + x.at(i);
      if (k < 0 or 100 * n <= k) continue;
      dp.at(i + 1).at(k) += dp.at(i).at(j);
    }
  }
  cout << dp.at(n).at(50 * n) - 1 << endl;
}