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)

S = X.map{|x| x - A}

def nCr(n, r)
  m = n - r
  if r > m
    ((r + 1)..n).inject(1, :*) / (1..m).inject(1, :*)
  else
    ((m + 1)..n).inject(1, :*) / (1..r).inject(1, :*)
  end
end

# zero
z = S.count(0)
if z > 0
  z = (1..z).inject(0) {|tot, r| tot + nCr(z, r)}
end

# minus
m = S.select{|x| x < 0}
hm = Hash.new(0)
(1..m.size).each do |r|
  m.combination(r) do |x|
    hm[- x.inject(:+)] += 1 
  end
end

# plus
q = S.select{|x| x > 0}
pm = Hash.new(0)
(1..q.size).each do |r|
  q.combination(r) do |x|
    pm[x.inject(:+)] += 1 
  end
end

tot = (hm.keys & pm.keys).inject(0){|tot, k| tot += hm[k] * pm[k]}
puts tot + z + (tot * z)