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<vector>
#include<fstream>
using namespace std;
#define ll long long
#define abs(n) (n < 0 ? -1 * (n) : (n))
int n, a;
vector<int> x(50);
vector<vector<ll>> m(51, vector<ll>(5002, -1));

ll ave(int i, int s)
{
    if(i == n) return 0;
    s += x[i];
    int sp0 = (s >= 0 ? s : abs(s) + 2500),
        smx = (s-x[i] >= 0 ? s-x[i] : abs(s-x[i]) + 2500);
    if(m[i+1][sp0] < 0) m[i+1][sp0] = ave(i+1, s);
    if(m[i+1][smx] < 0) m[i+1][smx] = ave(i+1, s-x[i]);
    ll k = m[i+1][sp0] + m[i+1][smx];
    //ll k = ave(i+1, s) + ave(i+1, s-x[i]);
    if(s == 0) k += 1;
    return k;
}

int main()
{
    cin >> n >> a;
    for(int i = 0; i < n; i++){
        cin >> x[i];
        x[i] -= a;
    }
    cout << ave(0, 0) << endl;
    return 0;
}