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;

namespace AtCoder
{
    class Program
    {
        static void Main(string[] args)
        {
            //[summary]C - 高橋君とカード / Tak and Cards
            var line = ReadLine();
            int n = line[0];
            int a = line[1];
            var x = ReadLine();

            var lib = new Library.Combination();
            long count = 0;

            for(int i = 1; i <= n; i++)
            {
                var combis = lib.AllCombinations<int>(x, i);

                foreach(var cards in combis)
                {
                    if (cards.Sum() == a * i)
                    {
                        count++;
                    }
                }
            }

            Console.WriteLine(count);
        }

        static List<int> ReadLine()
        {
            var line = Console.ReadLine();
            var array = line.Split(' ');

            return array.Select(x => int.Parse(x)).ToList();
        }
    }
}

namespace Library
{
    class Combination
    {
        public IEnumerable<List<T>> AllCombinations<T>(List<T> source, int n,
                                                       bool isAllowDupication = false)
        {
            //[summary]異なる要素から、n個を選ぶ組み合わせを列挙する
            //"isAllowDupication"を"true"にすると、重複を許可する
            //(重複を許可して、nを要素数と同じにすると、順列を列挙する)

            if (n == 1)
            {
                foreach (var s in source)
                {
                    yield return new List<T> { s };
                }

                yield break;
            }
            else
            {
                for (int i = 0; i < source.Count; i++)
                {
                    List<T> clone;

                    if (isAllowDupication)
                    {
                        //重複を許可する場合は、その要素のみを削除
                        clone = new List<T>(source);
                        clone.RemoveAt(i);
                    }
                    else
                    {
                        //重複を許可しない場合は、自分よりも前の要素はすべて削除
                        clone = new List<T>(source.Skip(i + 1));
                    }

                    //再帰処理
                    var items = AllCombinations(clone, n - 1, isAllowDupication);

                    foreach (var item in items)
                    {
                        item.Insert(0, source[i]);
                        yield return item;
                    }
                }
            }
        }
    }
}