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 <bits/stdc++.h>
#define _GLIBCXX_DEBUG
#define rep(i,n) for(int i=0;i<n;++i)
#define rrep(i,n) for(int i=n-1;i>=0;--i)
#define reps(i,a,b) for(int i=a;i<=b;++i)
#define fi first
#define se second
#define pb push_back
#define eb emplace_back
#define sz(x) (int)(x).size()
#define all(v) v.begin(),v.end()
#define rall(v) v.rbegin(),v.rend()
using namespace std;
using ll = long long;
using P = pair<int, int>;
template<typename T> inline bool chmax(T &a,T b){if(a<b){a=b;return 1;}return 0;}
template<typename T> inline bool chmin(T &a,T b){if(b<a){a=b;return 1;}return 0;}
template<typename T> vector<T> make_vec(size_t a){return vector<T>(a);}
template<typename T,typename... Ts>
auto make_vec(size_t a,Ts... ts){return vector<decltype(make_vec<T>(ts...))>(a,make_vec<T>(ts...));}
template<typename T,typename U,typename... V>
typename enable_if<is_same<T,U>::value>::type fill_v(U &u,const V... v){u=U(v...);}
template<typename T,typename U,typename... V>
typename enable_if<!is_same<T,U>::value>::type fill_v(U &u,const V... v){for(auto &e:u)fill_v<T>(e,v...);}

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    int n, a;
    cin >> n >> a;
    vector<int> x(n);
    rep(i, n) cin >> x[i];
    rep(i, n) x[i] -= a;
    vector<ll> dp(10000);
    dp[5000] = 1;
    rep(i, n) {
        vector<ll> pre(10000);
        swap(pre, dp);
        rep(j, 10000) {
            dp[j] += pre[j];
            int nj = j + x[i];
            if (0 <= nj && nj < 10000) dp[nj] += pre[j];
        }
    }
    cout << dp[5000] - 1 << endl;
}