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 (Clang 3.8.0) (judge verdict: Accepted)

#include <stdio.h>
typedef long long ll;

ll n, a;
ll x[55];
ll dp[55][55][55 * 55];
ll solve(ll sum, ll card, ll i);

int main()
{
	scanf("%lld %lld", &n, &a);
	for (ll i = 0; i < n; i++) {
		scanf("%lld", &x[i]);
	}

	for (ll i = 0; i < 55; i++) {
		for (ll j = 0; j < 55; j++) {
			for (ll k = 0; k < 55 * 55; k++) {
				dp[i][j][k] = -1;
			}
		}
	}

	printf("%lld\n", solve(0, 0, 0));

	return 0;
}

ll solve(ll sum, ll card, ll i)
{
	if (dp[i][card][sum] != -1) {
		return dp[i][card][sum];
	}

	if (i >= n && sum == card * a && card > 0) {
		return dp[i][card][sum] = 1;
	} else if (i >= n) {
		return dp[i][card][sum] = 0;
	}

	return dp[i][card][sum] = solve(sum + x[i], card + 1, i + 1) + 
		solve(sum, card, i + 1);
}