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 <cstdio>
#include <cstdlib>
#include <iostream>
#include <algorithm>
#include <cstring>
#include <vector>
#include <map>
#include <cmath>
#include <numeric>
 
#define rep(i,n) for(int i = 0;i < (n);i++)
#define all(x) (x).begin(),(x).end()
#define pb(x) push_back(x)
#define mp(a,b) make_pair(a,b)
#define maxelem(v) max_element(all(v))
#define minelem(v) min_element(all(v))
using namespace std;

typedef long long int ll;
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef vector<ll> vl;
typedef vector<pair<int, int> > vpi;
typedef vector<pair<ll, ll> > vpl;


ll dp[55][5555]={0};

int main(){
  cin.tie(0);
  ios::sync_with_stdio(false);
	int n, a;
  cin >> n >> a;
  vi y(n);
  rep(i,n){
    int x;
    cin >> x;
    y[i] = x - a;
  }
  sort(all(y));
  int mini = abs(y[0]);
  int maxi = abs(y[y.size()-1]);
  rep(i,n){
    if(i == 0){
      dp[0][n*mini+y[0]] = 1;
      continue;
    }
    dp[i][n*mini+y[i]]++;
    rep(j,n*(maxi+mini+1)){
      if(dp[i-1][j] != 0 || j==n*mini ){
        dp[i][j] += dp[i-1][j];
        dp[i][j+y[i]] += dp[i-1][j];
      }
    }
  }
  #ifdef debug
  rep(j,n*(maxi+mini+1)){
    if(j == n*mini) cout << "*" << " ";
    else cout << "  ";
  }
  cout << endl;
  rep(i,n){
    rep(j,n*(maxi+mini+1)){
      cout << dp[i][j] << " ";
    }
    cout << endl;
  }
  #endif
  cout << dp[n-1][n*mini] << endl;
	return 0;
}