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.chomp.split(' ').map(&:to_i)
@xs = gets.chomp.split(' ').map(&:to_i)
@memo = {}

class Array
  def sum
    self.reduce { |acc, x| acc + x }
  end
end

def solve(i, n, a)
  if @memo[[i, n, a]]
    return @memo[[i, n, a]]
  end

  if n == 1
    @memo[[i, n, a]] = @xs.slice(i, @N).select { |x| x == a }.count
    return @memo[[i, n, a]]
  end

  if @xs.size - i < n
    return 0
  end

  @memo[[i, n, a]] = [
    solve(i + 1, n - 1, (a - @xs[i])), # xs[i]を使ったとき
    solve(i + 1, n, a) # xs[i]を使わないとき
  ].sum
  @memo[[i, n, a]]
end

puts [*1..@N].map { |n| solve(0, n, A * n) }.sum