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, *ary = $stdin.read.split.map(&:to_i)
ary[0, 0] = nil

memoed_kosu = Array.new(n + 1){ Array.new(n + 1) {Array.new(a*n + 1)}};0

cal_memo = lambda do |j, k, a_sum|
  memoed_kosu[j][k][a_sum] ||= if k == 1
    ary[1, j].count(a_sum)
  elsif j == k # なくても多分良い
    ary[1, j].reduce(&:+) == a_sum ? 1 : 0
  elsif j < k # j個の中からk個を選ぶので
    0
  elsif a_sum - ary[j] >= 0
    cal_memo.(j-1, k, a_sum) + cal_memo.(j-1, k-1, a_sum - ary[j])
  else
    cal_memo.(j-1, k, a_sum)
  end
end
puts (1..n).reduce(0){|sum, i| sum + cal_memo.(n, i, a*i)}