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)

def f(x, n, k, count, wa, a)
  if count == n then
    return (wa == k*a ? 1 : 0)
  else
    ans = 0
    ans += f(x, n, k+1, count+1, wa+x[count], a)
    ans += f(x, n, k, count+1, wa, a)
    return ans
  end
end

def f2(dp, x, j, k, s)
  return 1 if j == 0 && k == 0 && s == 0
  return dp[j][k][s] if dp[j][k][s] != -1
  if j >= 1 && s < x[j-1] then
    dp[j][k][s] = f2(dp,x,j-1,k,s)#dp[j-1][k][s]
  elsif j >= 1 && k >= 1 && s >= x[j-1] then
    dp[j][k][s] = f2(dp,x,j-1,k,s) + f2(dp,x,j-1,k-1,s-x[j-1])#dp[j-1][k][s] + dp[j-1][k-1][s-x[j-1]]
  else
    dp[j][k][s] = 0
  end
  return dp[j][k][s]
end

N, A = gets.split.map(&:to_i)
x = gets.split.map(&:to_i)

#puts f(x, N, 0, 0, 0, A)-1 if N <= 20

X = x.max < A ? A : x.max
dp = Array.new(N+1){Array.new(N+1){Array.new(N*X+1,-1)}}
dp[0][0][0] = 1

ans = 0
(1..N).each do |k|
  ans += f2(dp,x,N,k,k*A)
end

#p dp
puts ans