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>
#include <math.h>

int cmpnum(const void * n1, const void * n2) {
	if (*(long *)n1 > *(long *)n2)
	{
		return -1;
	}
	else if (*(long *)n1 < *(long *)n2)
	{
		return 1;
	}
	else
	{
		return 0;
	}
}

int main(void) {

  long n,a;
  scanf("%ld %ld", &n, &a);
  long x[n];
  long plus = 0, minus = 0, zero = 0;
  for (long i = 0; i < n; i++) {
    scanf("%ld", &x[i]);
    x[i] -= a;
    if (x[i] > 0) {
      plus++;
    } else if (x[i] < 0) {
      minus++;
    } else {
      zero++;
    }
  }
  qsort(x, n, sizeof(long), cmpnum);
  long plus_sum[2500+1],minus_sum[2500+1],tmp[2500+1];
  plus_sum[0] = 1;
  minus_sum[0] = 1;
  for (long i = 1; i <= 2500; i++) {
    plus_sum[i] = 0;
    minus_sum[i] = 0;
  }
  for (long i = 0; i < plus; i++) {
    for (long j = 0; j <= 2500; j++) {
      tmp[j] = 0;
    }
    for (long j = 0; x[i]+j <= 2500; j++) {
      tmp[x[i]+j] = plus_sum[j];
    }
    for (long j = 0; j <= 2500; j++) {
      plus_sum[j] += tmp[j];
    }
  }
  for (long i = plus+zero; i < n; i++) {
    for (long j = 0; j <= 2500; j++) {
      tmp[j] = 0;
    }
    for (long j = 0; -x[i]+j <= 2500; j++) {
      tmp[-x[i]+j] = minus_sum[j];
    }
    for (long j = 0; j <= 2500; j++) {
      minus_sum[j] += tmp[j];
    }
  }
  long ans = 0;
  for (long i = 0; i <= 2500; i++) {
    ans += plus_sum[i]*minus_sum[i];
  }
  ans *= pow(2,zero);
  ans--;
  printf("%ld\n", ans);

  return 0;
}