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.Collections;
using System.Text;
using System.Threading.Tasks;
using static System.Math;
using static System.Console;
using System.Collections.Generic;
using System.Linq;
using System;
using System.Numerics;

class Program
{
    #region Reader
    //static Vector2Int ReadVec2Int
    //{
    //    get
    //    {
    //        string[] s = Console.ReadLine().Split(' ');
    //        return new Vector2Int(int.Parse(s[0]), int.Parse(s[1]));
    //    }
    //}
    //static Vector2Int[] ReadVec2Ints(int n)
    //{
    //    List<Vector2Int> list = new List<Vector2Int>();
    //    for (int i = 0; i < n; ++i)
    //    {
    //        var ab = ReadInts;
    //        list.Add(new Vector2Int(ab[0], ab[1]));
    //    }
    //    return list.ToArray();
    //}
    //static HashSet<Vector2Int> ReadVec2IntHash(int n)
    //{
    //    HashSet<Vector2Int> hash = new HashSet<Vector2Int>();
    //    for (int i = 0; i < n; ++i)
    //    {
    //        var ab = ReadInts;
    //        hash.Add(new Vector2Int(ab[0], ab[1]));
    //    }
    //    return hash;
    //}
    static string ReadStr => Console.ReadLine();
    static string[] ReadStrs => Console.ReadLine().Split(' ');
    static int ReadInt => Convert.ToInt32(Console.ReadLine());
    static int[] ReadInts => Console.ReadLine().Split(' ').Select(s => Convert.ToInt32(s)).ToArray();
    static long ReadLong => Convert.ToInt64(Console.ReadLine());
    static long[] ReadLongs => Console.ReadLine().Split(' ').Select(s => Convert.ToInt64(s)).ToArray();
    static long[,] ReLongMatrix(int r, int c)
    {
        var mat = new long[r, c];
        for (int i = 0; i < r; ++i)
        {
            var x = ReadLongs;
            for (int j = 0; j < c; ++j)
            {
                mat[i, j] = x[j];
            }
        }
        return mat;
    }
    static Tuple<string, string>[] ReTupleStr(int n)
    {
        List<Tuple<string, string>> list = new List<Tuple<string, string>>();
        for (int i = 0; i < n; ++i)
        {
            var ab = ReadStrs;
            list.Add(new Tuple<string, string>(ab[0], ab[1]));
        }
        return list.ToArray();
    }
    static Tuple<long, long>[] ReTupleLong(int n)
    {
        List<Tuple<long, long>> list = new List<Tuple<long, long>>();
        for (int i = 0; i < n; ++i)
        {
            var ab = ReadLongs;
            list.Add(new Tuple<long, long>(ab[0], ab[1]));
        }
        return list.ToArray();
    }
    static Tuple<int, int>[] ReadTupleInt(int n)
    {
        List<Tuple<int, int>> list = new List<Tuple<int, int>>();
        for (int i = 0; i < n; ++i)
        {
            var ab = ReadInts;
            list.Add(new Tuple<int, int>(ab[0], ab[1]));
        }
        return list.ToArray();
    }
    static Tuple<double, double>[] ReTupleDouble(int n)
    {
        List<Tuple<double, double>> list = new List<Tuple<double, double>>();
        for (int i = 0; i < n; ++i)
        {
            var ab = ReadLongs;
            list.Add(new Tuple<double, double>(ab[0], ab[1]));
        }
        return list.ToArray();
    }
    static Dictionary<int, HashSet<int>> ReAdjacencyList(int N, int M)
    {
        Dictionary<int, HashSet<int>> dict = new Dictionary<int, HashSet<int>>();
        for (int i = 0; i <= N; ++i)
        {
            dict.Add(i, new HashSet<int>());
        }
        for (int i = 0; i < M; ++i)
        {
            var ab = ReadInts;
            dict[ab[0]].Add(ab[1]);
            dict[ab[1]].Add(ab[0]);
        }
        return dict;
    }
    #endregion
    static void Swap<T>(ref T a, ref T b)
    {
        T temp = a;
        a = b;
        b = temp;
    }
    static int mod = (int)Pow(10, 9) + 7;

    public static long GetInv(long a, long mod = 1000000007)
    {
        return PowerMod(a, mod - 2, mod);
    }
    public static long Combination(long a, long b, long mod = 1000000007)
    {
        if (b < 0 || b > a) return 0;
        if (b > a / 2) b = a - b;
        long numerator = Factorial(a, mod);
        long denominator = (Factorial(b, mod) * Factorial(a - b, mod)) % mod;

        return (numerator * GetInv(denominator, mod)) % mod;
    }
    public static long Factorial(long n, long mod = 1000000007)
    {
        if (n == 0) return 1;
        if (n == 1) return 1;
        long ans = n;
        for (int i = 2; i < n; i++)
        {
            ans = (ans * i) % mod;
        }

        return ans;
    }
    private static long PowerMod(long a, long n, long mod = 1000000007)
    {
        if (n == 0) return 1;
        if (n == 1) return a;

        long p = PowerMod(a, n / 2, mod);
        if (n % 2 == 1) return ((p * p) % mod * a) % mod;
        else return (p * p) % mod;
    }


    static void Main()
    {
        string S = ReadStr;
        int N = S.Length;
        short K = short.Parse(ReadLine());

        int cnt = (int)Combination(Max(N - 1, 0), K, mod) * (int)Pow(9, K);

        //最も次数が高い数字が異なる
        cnt += ((S[0] - '0') - 1) * ((int)Combination(N - 1, K - 1, mod)) * (int)Pow(9, K - 1);
        if (K == 1)
        {
            WriteLine(cnt + 1);
            return;
        }

        //同じとき、次に0ではない数を探す
        int index = -1;
        for (int i = 1; i < N; i++)
        {
            if (S[i] != '0')
            {
                index = i;
                break;
            }
        }
        if (K == 2)
        {
            if(index!=-1)cnt += Max(N - index - 1, 0) * 9 + (S[index] - '0');
            WriteLine(cnt);
            return;
        }
        //二番目が異なる
        if (index != -1)
        {
            cnt += ((S[index] - '0') - 1) * Max(N - index - 1, 0) * 9;
            cnt += (int)Combination(N - index - 1, 2, mod) * 9 * 9;
        }

        //同じとき
        int index2 = -1;
        if (index != -1)
        {
            for (int i = index + 1; i < N; i++)
            {
                if (S[i] != '0')
                {
                    index2 = i;
                    break;
                }
            }
        }

        if (index2 != -1)
        {
            cnt += Max(N - index2 - 1, 0) * 9 + (S[index2] - '0');
        }


        WriteLine(cnt);

    }

}