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)*a+(j))

int calc(int *a,int n,int64 *dp){
  int i,j;
  for(i=0;i<=50*n;i++) dp[i]=0;
  dp[0]=1;
  int sum=0;
  for(i=0;i<n;i++){
    for(j=sum;j>=0;j--){
      dp[j+a[i]]+=dp[j];
    }
    sum+=a[i];
  }
  return sum;
}

void run(void){
  int n,a;
  scanf("%d%d",&n,&a);
  int *minus=(int *)malloc(sizeof(int)*n);
  int *plus=(int *)malloc(sizeof(int)*n);
  int mlen=0;
  int plen=0;
  int eq=0;
  int i;
  for(i=0;i<n;i++){
    int x;
    scanf("%d",&x);
    if(x==a){
      eq++;
    } else if(x<a){
      minus[mlen++]=a-x;
    } else {
      plus[plen++]=x-a;
    }
  }
  int64 *minusDP=(int64 *)malloc(sizeof(int64)*(50*mlen+1));
  int64 *plusDP =(int64 *)malloc(sizeof(int64)*(50*mlen+1));
  int mUP=calc(minus,mlen,minusDP);
  int pUP=calc(plus,plen,plusDP);
  int64 ans=0;
  for(i=0;i<=mUP && i<=pUP;i++){
    ans+=minusDP[i]*plusDP[i];
  }
  ans=(ans<<eq)-1;
  printf("%lld\n",ans);
}

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