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> cards(N);
    int maxX = -1*(1<<30);
    for(int i=0; i<N; ++i){
        cin >> cards[i];
        maxX = max(maxX, cards[i]);
        cards[i] -= A;
    }
    const int M = N * maxX;
    vector<vector<long long> > DP(N+1, vector<long long>(2*M+1, 0));
    DP[0][M] = 1;
    for(int i=1; i<=N; ++i){
        for(int j=0; j<2*M+1; ++j){
            DP[i][j] += DP[i-1][j];
            if(j-cards[i-1]>=0 and j-cards[i-1]<=2*M){
                DP[i][j] += DP[i-1][j-cards[i-1]];
            }
        }
    }
    cout << DP[N][M] -1 << endl;
    return 0;
}