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 Ruby (2.3.3) (judge verdict: Accepted)

@n, @a = gets.chomp.split.map(&:to_i)
@arr = gets.chomp.split.map(&:to_i).sort
@count = 0

if @a<@arr[0]
  puts 0
  exit
end

# 方針を変えて最後まで探索することにする.
# 配列をもたせるとメモ化できないので、それまでの合計値ももたせる
# 更にそれだけじゃダメなので、今までに取った数ももたせる
@memo = Array.new(50+1).map{|x| Array.new(50+1).map{|x| Array.new(50*50+1, nil)}}
def func(i, j, s)
  if !(@memo[i][j][s].nil?)
    return @memo[i][j][s]
  end
  if i==@n
    # 平均をとる 整数で@aとなることに注意
    if (s.to_f/(j).to_f) == @a.to_f
      return 1
    else
      return 0
    end
  else
    ans = 0 # ?
    # 追加する場合としない場合
    ans += func(i+1, j+1, s+@arr[i]) + func(i+1, j, s)
    @memo[i][j][s] = ans
    return ans
  end
end

puts func(0, 0, 0)