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)


def count_zero_sum_combination(cards, terms_num,
                               min_index = 0, sum = 0, hash = Hash.new(nil))
  return sum == 0 ? 1 : 0 if terms_num == 0
  return 0                if cards.size <= min_index
  hash_data = hash[[terms_num, min_index, sum]]
  return hash_data if hash_data != nil
  count = 0
  min_index.upto(cards.size - 1) do |index|
    count += count_zero_sum_combination(cards, terms_num - 1,
                                        index + 1, sum + cards[index], hash)
  end
  hash[[terms_num, min_index, sum]] = count
  return count
end

Cards_Num, Mean_Target = gets.strip.split(/\s+/).map{ |str| str.to_i }
Cards = gets.strip.split(/\s/).map{ |str| str.to_i - Mean_Target }.sort

answer = 0
Cards_Num.times do |i|
  answer += count_zero_sum_combination(Cards, i + 1)
end

puts answer