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: Accepted)

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

namespace C044
{
    class Program
    {
        static Dictionary<int,long> Calc(List<int> list)
        {
            Dictionary<int, long> dic = new Dictionary<int, long>();

            int size = list.Count;
            int sum = list.Sum();

            long[,] n=new long[size, sum+1];

            for (int i = 0; i < size; i++)
            {
                int a = list[i];
                if (i == 0)
                {
                    n[0, a] = 1;
                    n[0, 0] = 1;
                }
                else
                {
                    for (int j = 0; j < a; j++)
                    {
                        n[i, j] = n[i - 1, j];
                    }
                    for (int j = a; j <= sum; j++)
                    {
                        n[i, j] = n[i - 1, j] + n[i - 1, j - a];
                    }
                }
            }

            if (sum == 0)
            {
                dic.Add(0, 1);
            }
            else
            {
                for (int i = 0; i <= sum; i++)
                {
                    dic.Add(i, n[size - 1, i]);
                }
            }
            return dic;
        }

        static void Main(string[] args)
        {
            string[] s = Console.ReadLine().Split(' ');
            int N = int.Parse(s[0]);
            int A = int.Parse(s[1]);
            string[] s1 = Console.ReadLine().Split(' ');

            List<int> high = new List<int>();
            List<int> low = new List<int>();
            int counterA = 0;
            for (int i = 0; i < s1.Length; i++)
            {
                int x = int.Parse(s1[i]);
                if (x < A) low.Add(A - x);
                else if (x > A) high.Add(x - A);
                else counterA++;
            }

            low.Sort(); high.Sort();
            long nA = (long)Math.Pow(2, counterA);

            var dicL = Calc(low);
            var dicH = Calc(high);
            int m = Math.Min(dicL.Count, dicH.Count);
            long counter = 0;
            for(int i = 0; i < m; i++)
            {
                counter += dicL[i] * dicH[i] * nA;
            }
            Console.WriteLine(counter-1);
        }
    }
}