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 math.BigInt

  def solve(N: Int, A: Int, xs: List[Int]): BigInt = {
    val ys = Array(0) ++ xs.map(_ - A)
    val X = math.max(xs.max, A)
    val dp = Array.ofDim[BigInt](N + 1, 2 * N * X + 1)

    dp(0)(N * X) = 1
    for { j <- 0 to N; t <- 0 to 2 * N * X } {
      if (j == 0 && t == N * X) {
        dp(j)(t) = 1
      } else if (j >= 1 && (t - ys(j) < 0 || t - ys(j) > 2 * N * X)) {
        dp(j)(t) = dp(j - 1)(t)
      } else if (j >= 1 && (t - ys(j) >= 0 && t - ys(j) <= 2 * N * X)) {
        dp(j)(t) = dp(j - 1)(t) + dp(j - 1)(t - ys(j))
      } else {
        dp(j)(t) = 0
      }
    }
    val combs = dp(N)(N * X) - 1
    combs
  }

  val sc = new java.util.Scanner(Console.in)
  val n = sc.nextInt()
  val a = sc.nextInt()
  val nums = List.fill(n)(sc.nextInt())
  val ans = solve(n, a, nums)
  println(ans)
}