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: Time Limit Exceeded)

#include <stdio.h>

int x[50], A, N;
long long int ans;

void select(int rest, int count, int limit, int i);
void swap(int *n1, int *n2) {int tmp = *n1; *n1 = *n2; *n2 = tmp;}

int main(void)
{
	int i, j;

	scanf("%d%d", &N, &A);
	for (i = 0; i < N; i++) {
		scanf("%d", &x[i]);
	}

	for (i = N - 1; i > 0; i--) {
		for (j = 0; j < i; j++) {
			if (x[j] > x[j + 1])
				swap(&x[j], &x[j + 1]);
		}
	}

	for (i = 1; i <= N; i++) {
		select(A * i, 0, i, 0);
	}

	printf("%lld\n", ans);

	return 0;
}

void select(int rest, int count, int limit, int i)
{
	if (N < i) {
		return;
	} else if (rest < 0) {
		return;
	} else if (rest == 0 && limit == count) {
		ans++;
		return;
	} else {
		if (rest >= (limit - count) * x[i + 1]) {
			select(rest, count, limit, i + 1);
		}
		if (limit >= count + 1) {
			select(rest - x[i], count + 1, limit, i + 1);
		}
	}
}