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: Runtime Error)

N, A = gets.chomp.split(" ").map(&:to_i)
x = gets.chomp.split(" ").map(&:to_i)
K_MAX = x.max*N + 1
dp = Array.new(N+1){Array.new(N+1){Array.new(K_MAX, 0)}}
dp[0][0][0] = 1
(N+1).times do |i|
  (N+1).times do |j|
    K_MAX.times do |k|
      next if i < 1
      if k < x[i-1] then
        dp[i][j][k] = dp[i-1][j][k]
      elsif k >= 1
        dp[i][j][k] = dp[i-1][j][k] + dp[i-1][j-1][k - x[i-1]]
      end
    end
  end
end

ret = 0
(1..N).each do |j|
  ret += dp[N][j][j*A]
end
puts ret