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 (Clang 3.8.0) (judge verdict: Accepted)

#include<iostream>
#include<vector>
#include<algorithm>

#define REP(i,s,n) for(int i=s;i<n;++i)
#define rep(i,n) REP(i,0,n)

using namespace std;

typedef long long ll;

int N,A,x[51];

int main() {
  cin >> N >> A;
  rep(i,N) cin >> x[i];
  ll sum = 0;
  rep(_,N) {
    ll d = _ + 1;
    ll MAX = 50 * d + 1;
    vector<vector<ll> > dp(d+1,vector<ll>(MAX,0));
    dp[0][0] = 1;
    rep(i,N) {
      for(int j=d-1;j>=0;--j) {
	rep(k,MAX) {
	  if( dp[j][k] == 0 ) continue;
	  if( j + 1 <= d && k + x[i] <= d * A ) {
	    dp[j+1][k+x[i]] += dp[j][k];
	  }
	}
      }
    }
    sum += dp[d][d*A];
  }
  cout << sum << endl;
  return 0;
}