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 Rust (1.42.0) (judge verdict: Accepted)

fn solve() {
    let (n, a): (usize, usize) = (read::<usize>(), read::<usize>());
    let mut x: Vec<usize> = (0..n).map(|_| read::<usize>()).collect();
    // [index][sum][number]
    let mut dp = vec![vec![vec![0; n + 1]; a * n + 1]; n + 1];
    dp[0][0][0] = 1;
    for i in 1..n + 1 {
        for s in 0..a * n + 1 {
            for j in 0..i + 1 {
                if j > 0 {
                    match s.checked_sub(x[i - 1]) {
                        Some(subed) => {
                            dp[i][s][j] = dp[i - 1][subed][j - 1] + dp[i - 1][s][j];
                        }
                        None => dp[i][s][j] = dp[i - 1][s][j],
                    }
                } else {
                    dp[i][s][j] = dp[i - 1][s][j];
                }
            }
        }
    }
    let mut res: usize = 0;
    for i in 1..n + 1 {
        res += dp[n][i * a][i];
    }
    // println!("{:?}", dp);
    println!("{}", res);
}

fn main() {
    let stack_size = 104_857_600;
    let thd = std::thread::Builder::new().stack_size(stack_size);
    thd.spawn(|| solve()).unwrap().join().unwrap();
}

// =========
#[allow(unused_imports)]
use std::cmp::{max, min, Reverse};
#[allow(unused_imports)]
use std::collections::{BinaryHeap, HashMap, HashSet};
#[allow(unused_imports)]
use std::process::exit;

#[allow(dead_code)]
const MOD: usize = 1000000007;

fn read<T: std::str::FromStr>() -> T {
    use std::io::Read;
    let stdin = std::io::stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

// =========