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 <utility>
#include <tuple>
#include <vector>
#include <string>
#include <map>
#include <unordered_map>
#include <set>
#include <unordered_set>
#include <algorithm>
#include <functional>
#include <climits>
#include <numeric>
#include <queue>
#include <cmath>
#include <iomanip>
#include <array>
#include <string>
#include <stack>
#include <cassert>
#include <memory>

int main() {
	int n, a; std::cin >> n >> a;
	std::vector<int> cards(n); for (auto& c : cards) std::cin >> c;
	std::vector<std::vector<long long int>> pattern(n + 1, std::vector<long long int>(a * n + 1, 0));
	pattern[0][0] = 1;
	for (const auto c : cards) {
		for (int i = pattern.size() - 2; i >= 0; --i) {
			for (auto j = 0; j + c < pattern[i].size(); ++j) if (pattern[i][j] != 0) {
				pattern[i + 1][j + c] += pattern[i][j];
			}
		}
	}
	long long int result{ 0 };
	for (auto i = 1; i < pattern.size(); ++i) {
		result += pattern[i][i * a];
	}
	std::cout << result << '\n';
}