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 static System.Math;

namespace CSharp
{
    class Program
    {
        static long Solve(int[] X, long?[,,] memo, int no, int count, int sum)
        {
            if (no == X.Length || sum <= 0)
            {
                return sum == 0 && count == 0 ? 1 : 0;
            }
            if (count <= 0)
            {
                return 0;
            }
            if (memo[no, count, sum] != null)
            {
                return memo[no, count, sum].GetValueOrDefault();
            }

            return (memo[no, count, sum] = Solve(X, memo, no + 1, count, sum)
                                           + Solve(X, memo, no + 1, count - 1, sum - X[no])).GetValueOrDefault(); ;
        }

        static void Main(string[] args)
        {
            int n, a;
            ReadInt(out n, out a);
            int[] X = ReadIntArray();

            long ans = 0;
            long?[,,] memo = new long?[55, 55, 3000];
            for (int i = 1; i <= n; i++)
            {
                ans += Solve(X, memo, 0, i, i * a);
            }

            Console.WriteLine(ans);
        }

        #region Helpers

        static int[] ReadIntArray() => Console.ReadLine().Split().Select(s => int.Parse(s)).ToArray();
        static long[] ReadLongArray() => Console.ReadLine().Split().Select(s => long.Parse(s)).ToArray();

        static string ReadString() => Console.ReadLine();
        static int ReadInt() => int.Parse(Console.ReadLine());
        static long ReadLong() => long.Parse(Console.ReadLine());

        static void ReadInt(out int first)
        {
            var array = ReadIntArray();
            first = array[0];
        }

        static void ReadInt(out int first, out int second)
        {
            var array = ReadIntArray();
            first = array[0];
            second = array[1];
        }

        static void ReadInt(out int first, out int second, out int third)
        {
            var array = ReadIntArray();
            first = array[0];
            second = array[1];
            third = array[2];
        }

        static void ReadInt(out int first, out int second, out int third, out int fourth)
        {
            var array = ReadIntArray();
            first = array[0];
            second = array[1];
            third = array[2];
            fourth = array[3];
        }

        static void ReadLong(out long first)
        {
            var array = ReadLongArray();
            first = array[0];
        }

        static void ReadLong(out long first, out long second)
        {
            var array = ReadLongArray();
            first = array[0];
            second = array[1];
        }

        static void ReadLong(out long first, out long second, out long third)
        {
            var array = ReadLongArray();
            first = array[0];
            second = array[1];
            third = array[2];
        }

        static void ReadLong(out long first, out long second, out long third, out long fourth)
        {
            var array = ReadLongArray();
            first = array[0];
            second = array[1];
            third = array[2];
            fourth = array[3];
        }

        #endregion
    }
}