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: Accepted)

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;
const int maxn=1e6+5;
vector<ll> V;
ll pre[maxn];
ll dp[55][5005];
int main() {
    dp[0][0]=1;
    int n,a;cin>>n>>a;
    for(int i=1;i<=n;i++) cin>>pre[i];
    for(int i=1;i<=n;i++) {
    	for(int j=i;j>=1;j--) {
    		for(int k=5000;k>=pre[i];k--) {
    			dp[j][k]+=dp[j-1][k-pre[i]];
    		}
    	}
    }
    ll ans=0;
    for(int j=1;j<=n;j++) {
	    for(int i=0;i<=5000;i++) {
	    	if(a*j==i) ans+=dp[j][i];
	    }
    }
    cout<<ans<<endl;
    return 0;
}