Problem:
Score : 500 points

Problem Statement

Find the number of integers between 1 and N (inclusive) that contains exactly K non-zero digits when written in base ten.

Constraints

1 \leq N < 10^{100}

1 \leq K \leq 3

Input

Input is given from Standard Input in the following format:

N
K

Output

Print the count.

Sample Input 1

100
1

Sample Output 1

19

The following 19 integers satisfy the condition:

1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,100

Sample Input 2

25
2

Sample Output 2

14

The following 14 integers satisfy the condition:

11,12,13,14,15,16,17,18,19,21,22,23,24,25

Sample Input 3

314159
2

Sample Output 3

937

Sample Input 4

9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
3

Sample Output 4

117879300

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::collections::BinaryHeap;
use std::collections::HashMap;
use std::collections::HashSet;
use std::collections::VecDeque;

macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
    ($next:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

const MOD: i64 = 1_000_000_007;

fn comb(n: u32, k: u32) -> u32 {
    if n < k {
        return 0;
    }

    let mut c = 1;
    let m = if n - k > k { n - k } else { k };
    for i in m + 1..n + 1 {
        c *= i;
    }
    for i in 1..(n - m + 1) {
        c /= i;
    }
    return c;
}

fn recursive(i: u32, s: Vec<char>, k: u32) -> u32 {
    if k <= 0 {
        return 1;
    }
    if i >= s.len() as u32 {
        return 0;
    }
    let mut c = 0;
    let n = s[i as usize] as u32 - '0' as u32;
    if s.len() as u32 - i - 1 >= k && n != 0 { // 先頭を０とした場合
        c += comb(s.len() as u32 - i - 1, k) as u32 * (9 as u32).pow(k);
    }
    if n > 1 { // 先頭から１引いた場合
        c += (n - 1) * comb(s.len() as u32 - i - 1, k - 1) as u32 * (9 as u32).pow(k - 1);
    }
    if n != 0 {
        c += recursive(i + 1, s, k - 1);
    } else {
        c += recursive(i + 1, s, k);
    }
    return c;
}

fn main() {
    input! {
        s: chars,
        k: u32,
    }

    println!("{}", recursive(0, s, k));
}