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)

N, A = gets.split(/ /).collect{|i| i.to_i}


list_s = []
count_j = 0
list_l = []

gets.split(/ /).collect{|x|
	x = x.to_i
	if x == A
		count_j += 1
	elsif x < A
		list_s << (A - x)
	elsif
		list_l << (x - A)
	end
}

def make_patterns(list)
	results = {0 => 1}
	
	list.each{|x|
		new_results = {}
		results.each{|sum, count|
			if new_results[sum]
				new_results[sum] += count
			else
				new_results[sum] = count
			end
			
			new_sum = sum + x
			if new_results[new_sum]
				new_results[new_sum] += count
			else
				new_results[new_sum] = count
			end
		}
		results = new_results
	}
	
	results
end

results_s = make_patterns(list_s)
results_l = make_patterns(list_l)

pattern_count_j = 2 ** count_j
count = pattern_count_j - 1
results_s.each{|sum_s, count_s|
	if sum_s > 0 && count_l = results_l[sum_s]
		count += count_s * count_l * pattern_count_j
	end
}

p count