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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <iostream>
#include <iomanip>
#include <cmath>
#include <functional>
#include <algorithm>
#include <numeric>
#include <vector>
#include <list>
#include <queue>
#include <stack>
#include <deque>
#include <map>
#include <set>
#include <unordered_map>
#include <unordered_set>
#include <boost/range/irange.hpp>
#include <boost/container/static_vector.hpp>
#include <boost/utility/string_ref.hpp>
#include <boost/multiprecision/cpp_int.hpp>
#include <boost/math/common_factor_rt.hpp>

class input
{
    std::istream& cin;

public:
    input() : cin(std::cin)
    {
        std::cin.tie(nullptr);
        std::ios::sync_with_stdio(false);
    }

    template <class T>
    inline input& operator>>(T& var)
    {
        cin >> var;
        return *this;
    }
    template <class T>
    inline input& operator,(T& var)
    {
        cin >> var;
        return *this;
    }

    template <class T>
    inline T read()
    {
        T var;
        cin >> var;
        return var;
    }

    template <class InputIterator>
    inline void read(InputIterator head, std::size_t n)
    {
        for (std::size_t i = 0; i < n; ++i) {
            cin >> *head;
            ++head;
        }
    }

    template <class T>
    inline std::vector<T> read(std::size_t n)
    {
        std::vector<T> res;
        res.reserve(n);

        for (std::size_t i = 0; i < n; ++i) {
            T tmp;
            cin >> tmp;
            res.emplace_back(tmp);
        }
        return res;
    }

    template <class T, class F>
    inline void online(std::size_t n, F&& f)
    {
        T var;
        for (std::size_t i = 0; i < n; ++i) {
            cin >> var;
            f(var);
        }
    }
};
template <class T, std::size_t n>
std::istream& operator>>(std::istream& in, std::array<T, n>& var)
{
    for (std::size_t i = 0; i < n; ++i) {
        in >> var[i];
    }
    return in;
}

class output
{
    std::ostream& cout;

    template <class T, class... Args>
    void impl(T&& v, Args&&... args)
    {
        cout << std::forward<T>(v) << ' ';
        impl(std::forward<Args>(args)...);
    }
    template <class T>
    void impl(T&& v)
    {
        cout << std::forward<T>(v);
    }

public:
    output() : cout(std::cout)
    {
        cout << std::fixed << std::setprecision(15);
    }

    template <class... Args>
    void operator()(Args&&... args)
    {
        impl(std::forward<Args>(args)...);
        cout << '\n';
    }
    template <class... Args>
    void noreturn(Args&&... args)
    {
        impl(std::forward<Args>(args)...);
    }

    template <class T>
    void print(const T& container)
    {
        bool first = true;
        for (auto&& x : container) {
            if (!first) {
                cout << ' ';
            }
            cout << x;
            first = false;
        }
        cout << '\n';
    }
};

std::uint64_t comb(std::uint64_t n, std::uint64_t k)
{
    if (n < k) return 0;
    if (k == 0) return 1;
    return comb(n - 1, k - 1) * n / k;
}

std::uint64_t f(boost::string_ref s, int k)
{
    if (s.size() == 0 || k == 0) return 0;

    int x = s[0] - '0';
    s.remove_prefix(1);
    if (x) {
        std::uint64_t res = 0;
        res += (comb(s.size(), k) * std::pow(9, k));
        res += (x - 1) * (comb(s.size(), k - 1) * std::pow(9, k - 1));
        if (k == 1) res += 1;
        return f(s, k - 1) + res;
    }
    return f(s, k);
}

void prog()
{
    input in;
    output out;
    std::string n;
    int k;
    in >> n, k;
    out(f(n, k));
}

int main()
{
    prog();
    return 0;
}