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 Lua (5.3.2) (judge verdict: Accepted)

local n, a = io.read("*n", "*n")
local t = {}
for i = 1, n * 50 * n do
  t[i] = 0
end
for i = 1, n do
  local x = io.read("*n")
  for j = i - 1, 1, -1 do
    for k = n * 50 - x, 1, -1 do
      if 0 < t[n * 50 * (j - 1) + k] then
        t[n * 50 * j + k + x] = t[n * 50 * j + k + x] + t[n * 50 * (j - 1) + k]
      end
    end
  end
  t[x] = t[x] + 1
end
local cnt = 0
for i = 1, n do
  cnt = cnt + t[n * 50 * (i - 1) + a * i]
end
print(cnt)