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)

import java.util.Scanner

object Main {
  def main(args: Array[String]): Unit = {
    val in = new Scanner(System.in)
    val N = in.nextInt()
    val A = in.nextInt()
    val x = Array.ofDim[Int](N)
    x.indices foreach (i => x(i) = in.nextInt())
    System.out.println(solve(N, A, x))
  }

  def solve(N: Int, A: Int, x: Array[Int]): Long = {
    val dp = Array.ofDim[Long](N + 1, 100 * 50 + 1) // dp(i)(Aとの差の累積和, (-50, 50) を (0, 100)で表現している)
    dp(0)(2500) = 1 // 1まいも選んでないケースを１で初期化
    for {
      i <- 0 until N
      j <- 0 to 100 * 50
    } {
      val d = j + x(i) - A
      if (d >= 0 && d <= 5000) dp(i + 1)(d) += dp(i)(j) // i番目を選んだ場合
      dp(i + 1)(j) += dp(i)(j) // i番目を選ばなかった
    }
    dp(N)(2500) - 1 // １枚も選んでいないケースを省く
  }
}