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: Time Limit Exceeded)

n, a = gets.split.map(&:to_i)
x = gets.split.map(&:to_i)
sum = x.inject(:+)
dp = Array.new(n + 1) { Array.new(n + 1) { Array.new(sum + 1) { 0 } } }
dp[0][0][0] = 1
(1..n).each do |i|
  (0..n).each do |j|
    (0..sum).each do |k|
      dp[i][j][k] =
        if j >= 1 && k >= x[i - 1]
          dp[i - 1][j][k] + dp[i - 1][j - 1][k - x[i - 1]]
        else
          dp[i - 1][j][k]
        end
    end
  end
end
ans = 0
(1..n).each do |j|
  break if j * a > sum
  ans += dp[n][j][j * a]
end
puts ans