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)

$cache = {}

def dp(j, k, s)
  return 1 if j == 0 && k == 0 && s == 0
  return 0 if j == 0

  key = [j, k, s].join('-')
  return $cache[key] if $cache.key?(key)

  index = j - 1
  if $x[index] > s
    ret = dp(j - 1, k, s)
  elsif k > 0
    ret = dp(j - 1, k, s) + dp(j - 1, k - 1, s - $x[index])
  else
    ret = 0
  end

  $cache[key] = ret
  ret
end

def solve(n, a)
  (1..n).reduce(0) do |acc, i|
    acc + dp(n, i, i * a)
  end
end

def main
  stdin = STDIN.gets
  na = stdin.split(/\s/).map!(&:to_i)
  n = na[0]
  a = na[1]
  stdin = STDIN.gets
  $x = stdin.split(/\s/).map!(&:to_i)

  puts solve(n, a)
end

main