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# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace ConsoleApplication1
{
    class Cards
    {
        bool[] memo;
        internal int[] cards;

        int n, target;

        public Cards(int n, int a)
        {
            memo = new bool[1 << n];
            this.n = n;
            target = a;
        }

        public long Solve()
        {
            DFS(0, 0, 0, 0);

            long ret = memo.Where(m => m == true).Count();

            return ret;
        }

        private void DFS(int depth, int vol, int sum, int mask)
        {
            if (vol > 0)
            {
                double ave = (double)sum / vol;
                
                if (ave == target) memo[mask] = true;
            }

            if (depth == n) return;

            DFS(depth + 1, vol + 1, sum + cards[depth], mask | 1 << depth);
            DFS(depth + 1, vol, sum, mask);
        }
    }

    class Program
    {
        static void Main(string[] args)
        {
            string[] input = Console.ReadLine().Split(' ');
            int n = int.Parse(input[0]);
            int a = int.Parse(input[1]);

            Cards cs = new Cards(n, a);

            input = Console.ReadLine().Split(' ');

            cs.cards = Array.ConvertAll(input, int.Parse);

            Console.WriteLine(cs.Solve());
        }
    }
}