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 <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cmath>
#include <cstdio>
#include <iostream>
#include <iterator>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <utility>
#include <vector>
// }}}

using namespace std;

using ll = long long;
using ull = unsigned long long;
using ld = long double;

static constexpr int mod = (int)1e9 + 7;
static constexpr int inf = 100100100;
static constexpr ll linf = 1e18;
static constexpr double eps = 1e-9;
static constexpr double pi = 3.14159265359;

#define rep(i, n) for (ll i = 0; i < n; ++i)
#define rrep(i, n) for (ll i = n; i >= 0; --i)
#define all(c) begin(c), end(c)
#define rall(c) rbegin(c), rend(c)
#define pb push_back
#define ist insert
#define fst first
#define snd second


ll dp[51][51][2501];

int main() {
    // cin.tie(0);
    // ios_base::sync_with_stdio(false);
    ll N, A;
    cin >> N >> A;
    vector<ll> X;
    X.pb(0);  // dummy
    rep (i, N) {
        ll x;
        cin >> x;
        X.pb(x);
    }
    sort(all(X));
    dp[0][0][0] = 1;
    for (ll j = 0; j <= N; ++j) {
        for (ll k = 0; k <= N; ++k) {
            for (ll s = 0; s <= N * A; ++s) {
                if (j >= 1 && s < X[j]) {
                    dp[j][k][s] = dp[j - 1][k][s];
                } else if (j >= 1 && k >= 1 && s >= X[j]) {
                    dp[j][k][s] = dp[j - 1][k][s] + dp[j - 1][k - 1][s - X[j]];
                }
            }
        }
    }
    ll ans = 0;
    for (ll k = 1; k <= N; ++k) {
        ans += dp[N][k][k * A];
    }
    cout << ans << endl;
}