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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Runtime.CompilerServices;
using System.Text;
using System.Numerics;
using static Template;
using static System.Console;
using static System.Convert;
using static System.Math;
using Pi = Pair<int, int>;

class Solver
{
    public void Solve(Scanner sc)
    {
        string N;int K;
        sc.Make(out N, out K);
        //dp[i][j]:=i個,j?tight
        var dp = Create(K + 1, () => new int[2]);
        dp[0][1] = 1;
        for(var i = 0; i < N.Length; i++)
        {
            var newdp = Create(K + 1, () => new int[2]);
            for (var k = 0; k <= K; k++) newdp[k][0] = dp[k][0];
            for (var k = 0; k <= K; k++)
            {
                if (N[i] == '0')
                {
                    newdp[k][1] = dp[k][1];
                }
                else newdp[k][0] += dp[k][1];
            }
            for(var k = 0; k < K; k++)
            {
                newdp[k + 1][0] += dp[k][0] * 9 + dp[k][1] * Max(0,(N[i] - '1'));
            }
            if (N[i] != '0')
                for (var k = 0; k < K; k++)
                {
                    newdp[k + 1][1] += dp[k][1];
                }
            dp = newdp;
        }
        WriteLine(dp[K][0]+dp[K][1]);
    }
}

#region Template
public static class Template
{
    static void Main(string[] args)
    {
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver().Solve(new Scanner());
        Console.Out.Flush();
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmin<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) == 1) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmax<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) == -1) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T a, ref T b) { var t = b; b = a; a = t; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static T[] Create<T>(int n, Func<T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(); return rt; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static T[] Create<T>(int n, Func<int, T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(i); return rt; }
    public static void Fail<T>(T s) { Console.WriteLine(s); Console.Out.Close(); Environment.Exit(0); }
}

public class Scanner
{
    public string Str => Console.ReadLine().Trim();
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public int[] ArrInt => Str.Split(' ').Select(int.Parse).ToArray();
    public long[] ArrLong => Str.Split(' ').Select(long.Parse).ToArray();
    public char[][] Grid(int n) => Create(n, () => Str.ToCharArray());
    public int[] ArrInt1D(int n) => Create(n, () => Int);
    public long[] ArrLong1D(int n) => Create(n, () => Long);
    public int[][] ArrInt2D(int n) => Create(n, () => ArrInt);
    public long[][] ArrLong2D(int n) => Create(n, () => ArrLong);
    private Queue<string> q = new Queue<string>();
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public T Next<T>() { if (q.Count == 0) foreach (var item in Str.Split(' ')) q.Enqueue(item); return (T)Convert.ChangeType(q.Dequeue(), typeof(T)); }
    public void Make<T1>(out T1 v1) => v1 = Next<T1>();
    public void Make<T1, T2>(out T1 v1, out T2 v2) { v1 = Next<T1>(); v2 = Next<T2>(); }
    public void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3) { Make(out v1, out v2); v3 = Next<T3>(); }
    public void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4) { Make(out v1, out v2, out v3); v4 = Next<T4>(); }
    public void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5) { Make(out v1, out v2, out v3, out v4); v5 = Next<T5>(); }
    public void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6) { Make(out v1, out v2, out v3, out v4, out v5); v6 = Next<T6>(); }
    //public (T1, T2) Make<T1, T2>() { Make(out T1 v1, out T2 v2); return (v1, v2); }
    //public (T1, T2, T3) Make<T1, T2, T3>() { Make(out T1 v1, out T2 v2, out T3 v3); return (v1, v2, v3); }
    //public (T1, T2, T3, T4) Make<T1, T2, T3, T4>() { Make(out T1 v1, out T2 v2, out T3 v3, out T4 v4); return (v1, v2, v3, v4); }
}
public class Pair<T1, T2> : IComparable<Pair<T1, T2>>
{
    public T1 v1;
    public T2 v2;
    public Pair() { }
    public Pair(T1 v1, T2 v2)
    { this.v1 = v1; this.v2 = v2; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2> p)
    {
        var c = Comparer<T1>.Default.Compare(v1, p.v1);
        if (c == 0)
            c = Comparer<T2>.Default.Compare(v2, p.v2);
        return c;
    }
    public override string ToString() => $"{v1.ToString()} {v2.ToString()}";
    public void Deconstruct(out T1 a, out T2 b) { a = v1; b = v2; }
}

public class Pair<T1, T2, T3> : Pair<T1, T2>, IComparable<Pair<T1, T2, T3>>
{
    public T3 v3;
    public Pair() : base() { }
    public Pair(T1 v1, T2 v2, T3 v3) : base(v1, v2)
    { this.v3 = v3; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2, T3> p)
    {
        var c = base.CompareTo(p);
        if (c == 0)
            c = Comparer<T3>.Default.Compare(v3, p.v3);
        return c;
    }
    public override string ToString() => $"{base.ToString()} {v3.ToString()}";
    public void Deconstruct(out T1 a, out T2 b, out T3 c) { Deconstruct(out a, out b); c = v3; }
}
#endregion