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: Runtime Error)

#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#include<iostream>
#include<cstring>
#include<cstdint>
#include<cstddef>
using namespace std;
using i32 = int_fast32_t;
using i64 = int_fast64_t;
using usize = uint_fast64_t;
#define rep(i, n) for (usize i = 0; i < (usize)(n); i++)
#define all(a) (a).begin(),(a).end()
#define rall(a) (a).rbegin(),(a).rend()
using P = pair<i64,i64>;

int main(){
ios::sync_with_stdio(false);
std::cin.tie(nullptr);
i64 n,a;
cin >> n >> a;
usize x[n];
usize rec = 0;
rep(i,n){
    cin >> x[i];
    if(rec < x[i])
        rec = x[i];
}
usize dp[n + 1][n + 1][rec * n + 1];
memset(dp,0,sizeof(dp));
dp[0][0][0] = 1;
for(i32 i = 1; i <= n; i++){
    for(i32 j = 1; j <= n; j++){
        for(i32 k = 0; k <= rec * n; k++){
            if(k < x[i - 1])dp[i][j][k] = dp[i - 1][j][k];
            else if(k >= 1 && k >= x[i - 1])dp[i][j][k] = dp[i - 1][j][k] + dp[i - 1][j - 1][k - x[i - 1]];
        }
    }
}
i64 ans = 0;
for(i32 i = 1; i <= n; i++){
    ans += dp[n][i][i * a];
}
cout << ans << endl;
}