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

    let mut dp = vec![vec![vec![0; 5]; 2]; s.len() + 1];
    dp[0][0][0] = 1;
    for i in 1..s.len() + 1 {
        let n = s[i - 1] as usize - '0' as usize;
        for j in 0..k + 1 {
            // smallが確定していないものは現在位置まで入力とすべて同じ
            if n == 0 {
                // 現在のけたが0
                // smallが確定していなかったもの
                dp[i][0][j] += dp[i - 1][0][j];

                // smallが確定していたもの
                dp[i][1][j] += dp[i - 1][1][j]; // 0の場合
                dp[i][1][j + 1] += 9 * dp[i - 1][1][j]; // 0以外
            } else {
                // 0以外
                // smallが確定していなかったもの
                dp[i][0][j + 1] += dp[i - 1][0][j]; // nと同じ値
                dp[i][1][j] += dp[i - 1][0][j]; // 0でsmall確定
                dp[i][1][j + 1] += (n - 1) * dp[i - 1][0][j]; // 0以外でnより小さくsmall確定

                // smallが確定していたもの
                dp[i][1][j] += dp[i - 1][1][j]; // 0の場合
                dp[i][1][j + 1] += 9 * dp[i - 1][1][j]; // 0以外
            }
        }
    }

    //println!("{:?}", dp);
    println!("{}", dp[s.len()][0][k] + dp[s.len()][1][k]);
}