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.split.map(&:to_i)
x = [0] + gets.split.map(&:to_i)

X = (x + [A]).max

dp = Array.new(N + 1) do
  Array.new(N + 1) do
    Array.new(N * 50 + 1, 0)
  end
end

dp[0][0][0] = 1

(1..N).each do |j|
  (0..j).each do |k|
    (0..(N*X)).each do |s|
      dp[j][k][s] = case
                    when k == 0 && s == 0
                      1
                    when k == 0 || s == 0
                      0
                    when s < x[j]
                      dp[j-1][k][s]
                    when k >= 1 && s >= x[j]
                      dp[j-1][k][s] + dp[j-1][k-1][s-x[j]]
                    end
    end
  end
end

p (1..N).map { |k| dp[N][k][k*A] }.inject(:+)