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 Java8 (OpenJDK 1.8.0) (judge verdict: Runtime Error)

/*
 * 高橋君は、N 枚のカードを持っています。 i (1≤i≤N) 番目のカードには、整数 xi が書かれています。
 * 高橋君は、これらのカードの中から 1 枚以上を選び、 選んだカードに書かれた整数の平均をちょうど A にしたいと考えています。 そのようなカードの選び方が何通りあるか求めてください。
 */

import java.util.*;
	public class Main{
		public static void main(String[] args){
            Scanner sc = new Scanner(System.in);
            // 入力
            int n = Integer.parseInt(sc.next());
            int a = Integer.parseInt(sc.next());
            int[] nums = new int[n];
            int max = 1;

            for(int i=0; i<n; i++){
            	nums[i] = Integer.parseInt(sc.next());
            	if(max<nums[i]){
            		max = nums[i];
            	}
            }
            
            // a番目のカードまでのb枚を使って和をcにするのが何通りあるか = dp[a][b][c]
            long[][][] dp = new long[n+1][n+1][n*max+1];
            
            for(int i=0; i<n+1; i++){
            	for(int j=0; j<n*max+1; j++){
            		dp[0][i][j] = 0;
            		dp[i][0][j] = 0;
            	}
            }
            for(int i=0; i<n+1; i++){
            	dp[i][0][0] = 1;
            }
            
            for(int k=0; k<n*max+1; k++){
            	for(int j=1; j<n+1; j++){
            		for(int i=1; i<n+1; i++){
            			if(i<j){
            				dp[i][j][k] = 0;
            			}else{
            				dp[i][j][k] = dp[i-1][j][k];
            				if(k-nums[i-1] >= 0){
            					dp[i][j][k] += dp[i-1][j-1][k-nums[i-1]];
            				}
            				//System.out.println("dp[" + i + "][" + j +"][" + k + "] = " + dp[i][j][k]);
            			}
            		}
            	}
            }
            
            long ret = 0;
            
            for(int i=1; i<n+1; i++){
            	ret += dp[n][i][i*a];
            }
            
            // 出力
            System.out.println(ret);

            sc.close();
        }
    }