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 <iostream>
#include <string>
#include <algorithm>
#include <utility>
#include <vector>
#include <bitset>
#include <time.h>

using namespace std;
typedef long long  ll;
#define rep(i,s,n)for(int i = s;i<n;i++)
static const int M = 51;
int x[M];
ll dp[M][M][M*M];

int main()
{
	ll N, A; cin >> N >> A;
	rep(i, 0, N) {
		cin >> x[i];
	}
	rep(i, 0, N + 1) {
		rep(j, 0, N + 1) {
			rep(k, 0, N + 1) {
				dp[i][j][k] = 0;
			}
		}
	}
	dp[0][0][0] = 1;

	rep(i, 0, N) {
		rep(j, 0, N) {
			rep(k, 0, N*A) {
				if (k + x[i] <= N * A) {
					dp[i + 1][j + 1][k + x[i]] += dp[i][j][k];
				}
				dp[i + 1][j][k] += dp[i][j][k];
			}
		}
	}

	ll count = 0;
	rep(i, 1, N + 1) {
		count += dp[N][i][i*A];
	}
	cout << count << endl;
}