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() {
    ios::sync_with_stdio(false);
    cin.tie(0);

    int N, A; cin >> N >> A;
    int X = 0, y[N]; 
    for (int i = 0; i < N; ++i) { 
        int x; cin >> x; y[i] = x - A;
        X = max(X, x);
    }    

    long long dp[N + 1][2*X*N + 1];
    fill(dp[0], dp[N + 1], 0);

    for (int i = 1; i <= N; ++i) {
        dp[i][y[i - 1] + X*N] += 1;
        for (int j = 0; j < 2*X*N + 1; ++j) {
            dp[i][j] += dp[i - 1][j];
            int diff = j + y[i - 1];
            if (0 <= diff && diff <= 2*X*N)
                dp[i][diff] += dp[i - 1][j];
        }
    }

    cout << dp[N][X*N] << endl;
}