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: Accepted)

#class Array
#	def cumsum
#		r = 0
#		return self.map { |x| r+=x }
#	end
#end


x = $<.read.split.map(&:to_i)
N, A = x.shift(2)
x.map! { |y| A - y }.sort!

#idx = x.bsearch_index { |y| y >= 0 } || 0

#z = x[idx...N].reverse.cumsum.reverse || [0]
#z = [z[0]] * [idx,0].max + z  << 0

h = Hash.new(0)
h[0]=1

N.times do |i|
	hh = h.dup
	h.each do |k, v|
#		if x[i] + k + z[i] < 0
#			h.delete(x[i]+k)
#			next
#		p [x[i],k,hh[x[i]+k]]
		hh[x[i]+k] += v 
	end
	h = hh.dup
end
p h[0]-1