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, ave = gets.split.map(&:to_i)
cards  = gets.split.map(&:to_i)

plus  = []
minus = []
cnt0  = 0

cards.each do |i|
  x = i - ave
  if x == 0
    cnt0 += 1
  elsif x > 0
    plus  << x
  else
    minus << - x
  end
end

cntP = []
cntM = []

for i in 1 .. plus.size
  plus.combination(i).each do |a|
    x = a.inject(:+)
    if cntP[x] == nil
      cntP[x]  = 1
    else
      cntP[x] += 1
    end
  end
end

for i in 1 .. minus.size
  minus.combination(i).each do |a|
    x = a.inject(:+)
    if cntM[x] == nil
      cntM[x]  = 1
    else
      cntM[x] += 1
    end
  end
end

ans0 = 1

cntP.size.times do |i|
  if cntP[i] != nil && cntM[i] != nil
    ans0 += cntP[i] * cntM[i]
  end
end

ans1 = 1

ary0 = []
cnt0.times do |i|
  ary0 << i
end

for i in 1 .. cnt0
  ans1 += ary0.combination(i).size
end

puts [ans0 * ans1, ans0, ans1].max - 1