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 C (GCC 5.4.1) (judge verdict: Accepted)

#include<stdio.h>
#include<stdlib.h>

typedef long long int int64;

#define POS(i,j) ((i)*(2*50*n+1)+(j)+50*n)

void run(void){
  int n,a;
  scanf("%d%d",&n,&a);
  int64 *dp=(int64 *)calloc(2*50*n+1,sizeof(int64));
  int now=0;
  dp[POS(0,0)]=1;
  int l,r;
  l=r=0;
  int i,j;
  for(i=0;i<n;i++){
    int x;
    scanf("%d",&x);
    x-=a;
    int next=now^1;
    for(j=l;j<=r;j++) dp[POS(next,j)]=dp[POS(now,j)];
    for(j=l;j<=r;j++) dp[POS(next,j+x)]+=dp[POS(now,j)];
    if(x<0){
      l+=x;
    } else {
      r+=x;
    }
    now=next;
  }
  printf("%lld\n",dp[POS(now,0)]-1);
  return;
}

int main(void){
  run();
  return 0;
}