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: Time Limit Exceeded)

#include<bits/stdc++.h>

#define INF 1000000007
#define LINF 1000000000000000007

using namespace std;

typedef long long Int;
typedef pair<Int, Int> P;

Int gcd(Int x, Int y){
	if(y == 0)
		return x;
	return gcd(y, x % y);
}

Int lcm(Int x, Int y){
	return x * y / gcd(x, y);
}

bool sosuu(int x){
	bool b = 1;
	if(x == 1) b = 0;
	for(int i = 2; i * i <= x; i++){
		if(x % i == 0) {
			b = 0;
			break;
		}
	}
	return b;
}

Int n, a, x[52], ans = 0;
Int dp[51][51][2500] = {0};

Int dfs(int c, int i, int s){
	if(i == n){
		if(c > 0 && s > 0){
			if(c / s == a && c % s == 0){
				return 1;
			}
		}
		return 0;
	}
	if(dp[i][s][c] != 0) return dp[i][s][c];
	return dp[i][s][c] = dfs(c + x[i], i+1, s+1) + dfs(c, i+1, s);
}

int main(){
	cin >> n >> a;
	for(int i = 0; i < n; i++){
		cin >> x[i];
	}
	
	cout << dfs(0,0,0) << endl;
	
	return 0;
}