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 Scala (2.11.7) (judge verdict: Accepted)

object Main {
  def main(args:Array[String]) = {
    val sc = new java.util.Scanner(System.in)
    val n, a = sc.nextInt
    val x = List.fill(n)(sc.nextInt)

    val nx = (a :: x).max * n

    val dp = collection.mutable.ArrayBuffer.fill(n + 1, n + 1, nx + 1)(-1L)
    def dpFunc(j: Int, k: Int, s: Int): Long = {
    	if (dp(j)(k)(s) >= 0) return dp(j)(k)(s)

    	val ret = if (j == 0 && k == 0 && s == 0) 1L
    		else if (j > 0 && s < x(j - 1)) dpFunc(j - 1, k, s)
    		else if (j > 0 && k > 0 && s >= x(j - 1)) dpFunc(j - 1, k, s) + dpFunc(j - 1, k - 1, s - x(j - 1))
    		else 0L
    	dp(j)(k)(s) = ret
    	ret
    }

    val ans = (1 to n).map(k => dpFunc(n, k, k * a)).sum

    println(ans)

  }
}