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 extends App {

  import scala.io.StdIn._

  val Array(n, a) = readLine().split(' ').map(_.toInt)
  val values = readLine().split(' ').map(_.toInt).map(_ - a).toList
  def table : Map[Int, Long] ={
    def hoge(current: Map[Int, Long], values: List[Int]): Stream[Map[Int, Long]] = {

      def withValue(n: Int) = {
        val newKeys = current.keySet ++ current.keySet.map(_ + n)
        newKeys.map{k => k -> (current.getOrElse(k , 0L) + current.getOrElse(k - n, 0L))}.toMap
      }

      val next = values.headOption.map (withValue)

      val trail = next.map(hoge(_, values.tail)).getOrElse(Stream.Empty)
      current #:: trail
    }

    hoge(Map(0 -> 1), values)(n)
  }

  println(table(0) - 1)
}