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 (1.15.1) (judge verdict: Time Limit Exceeded)

fn main() {
    use std::collections::HashMap;

    let na: Vec<usize> = atcoder::read_multiple_num();
    let (n, a) = (na[0], na[1]);
    let x: Vec<u64> = atcoder::read_multiple_num();
    let mut dp = Vec::new();

    for i in 0..n {
        dp.push(HashMap::new());
        dp[i].insert((1 << i) as u64, x[i] as f64);
        if i != 0 {
            for (k, v) in dp[i - 1].clone() {
                dp[i].insert(k, v);
                dp[i].insert(k | (1 << i) as u64, v + x[i] as f64);
            }
        }
    }

    let count = dp[n - 1].clone().into_iter()
        .filter(|&(k, v)| {
            let num = k.count_ones();
            ((v / num as f64) - a as f64).abs() < std::f64::EPSILON
        }).count();

    println!("{}", count);
}

#[allow(dead_code)]
mod atcoder {
    pub fn read_single_num<T: ::std::str::FromStr>() -> T
        where <T as ::std::str::FromStr>::Err: ::std::fmt::Debug {
            let mut buffer = String::new();
            ::std::io::stdin().read_line(&mut buffer).unwrap();
            buffer.trim().parse().unwrap()
    }

    pub fn read_multiple_num<T: ::std::str::FromStr>() -> Vec<T>
        where <T as ::std::str::FromStr>::Err: ::std::fmt::Debug {
            let mut buffer = String::new();
            ::std::io::stdin().read_line(&mut buffer).unwrap();
            buffer.split_whitespace().map(|i| i.parse().unwrap() ).collect()
    }

    pub fn read_single_string() -> String {
            let mut buffer = String::new();
            ::std::io::stdin().read_line(&mut buffer).unwrap();
            buffer.trim().to_string()
    }

    pub fn read_multiple_string() -> Vec<String> {
            let mut buffer = String::new();
            ::std::io::stdin().read_line(&mut buffer).unwrap();
            buffer.split_whitespace().map(|s| s.to_string() ).collect()
    }
}