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)
x = gets.chomp.split.map(&:to_i)

dp = Array.new(N+1).map{Array.new(N+1).map{Array.new(50*N+1, 0)}}


# i = 0でjが1以上のものはないのでほかは0で良い
dp[0][0][0] = 1

1.upto(N) do |i|
  dp[i][0][0] = 1
  # 最大でi個までしか選ばない
  1.upto(i) do |j|
    1.upto(50*N) do |k|
      if k >= x[i-1]
        dp[i][j][k] = dp[i-1][j][k] + dp[i-1][j-1][k-x[i-1]]
      else
        dp[i][j][k] = dp[i-1][j][k]
      end
    end
  end
end

# ans = 0
# 1.upto(N) do |j|
#   1.upto(N*50) do |k|
#     if j * A == k
#       # p "j:#{j} k:#{k} dp:#{dp[N][j][k]}"
#       ans += dp[N][j][k]
#     end
#   end
# end
# puts ans

puts (1..N).map {|j| dp[N][j][j * A] }.inject(0,&:+)