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: Runtime Error)

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

int64_t test(int N, int A, int X[]);
int main(void) {
    char *p;
    char str[256];

    int *X;
    int N,A;
    int i,cnt=0;

    p = fgets(str, 256, stdin);
    for(i=0;i<256;i++){
        if(str[i] == ' ') {
            str[i] = '\0';
            N = atoi(p);
            A = atoi(&str[i+1]);
            break;
        }
    }
    
    X = malloc(sizeof(int) * N);
    p = fgets(str, 256, stdin);
    for(i=0;i<256;i++){
        if(str[i] == ' '){
            str[i] = '\0';
            X[cnt++] = atoi(p);
            p = &str[i+1];
        }
    }
    X[cnt] = atoi(p);
    
    printf("%lu\n", test(N,A,X));
    free(X);

    return 0;
}

static int64_t dp[51][5206]; 

int64_t test(int N, int A, int X[]) {
    int j,t;
    int64_t sum = 0;
    int tmax;
    int Xm = A;


    int *Y = malloc(sizeof(int) * (N+1));
    for(j=0;j<N;j++){
        Y[j] = X[j] - A;
        if(Xm < X[j]) Xm = X[j];
    }
    tmax = 2 * N * Xm;

    memset(dp, 0, sizeof(dp));
    dp[0][N * Xm] = 1;
    for(j=1;j<=N;j++){
        for(t=0;t<=tmax;t++){
            if((t-Y[j-1] < 0) || ((t-Y[j-1]) > tmax)){
                dp[j][t] = dp[j-1][t];
            } else if(0 <= (t-Y[j-1]) && (t-Y[j-1]) <= tmax){
                dp[j][t] = dp[j-1][t] + dp[j-1][t-Y[j-1]];
            }
        }
    }
    
    free(Y);

    return dp[N][N * Xm] - 1;
}