verify/math/binomial_coefficient_mod.test.cpp

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Last update: 2026-10-06 01:57:24+09:00
Problem: https://judge.yosupo.jp/problem/binomial_coefficient

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Code

#define PROBLEM "https://judge.yosupo.jp/problem/binomial_coefficient"

#include <cassert>
#include <cstdint>
#include "../../utilities/fast_io.hpp"
#include <vector>

#include "../../math/binomial_coefficient_mod.hpp"

namespace {

void test_against_pascal_triangle() {
    constexpr int maximum = 60;
    for (uint32_t modulus = 1; modulus <= 100; modulus++) {
        m1une::math::BinomialCoefficientMod combinations(modulus);
        assert(combinations.modulus() == modulus);

        std::vector<std::vector<uint32_t>> binom(maximum + 1);
        for (int n = 0; n <= maximum; n++) {
            binom[n].resize(n + 1);
            binom[n][0] = 1 % modulus;
            binom[n][n] = 1 % modulus;
            for (int k = 1; k < n; k++) {
                binom[n][k] = (binom[n - 1][k - 1] + binom[n - 1][k]) % modulus;
            }
            for (int k = 0; k <= n; k++) {
                assert(combinations.binom(n, k) == binom[n][k]);
                assert(combinations(n, k) == binom[n][k]);
            }
            assert(combinations.binom(n, uint64_t(n) + 1) == 0);
        }
    }
}

void test_large_arguments() {
    constexpr uint64_t n = 1000000000000000000ULL;
    for (uint32_t modulus : {1U, 2U, 8U, 9U, 12U, 999983U, 1000000U}) {
        m1une::math::ArbitraryModBinomialCoefficient combinations(modulus);
        assert(combinations(n, 0) == 1 % modulus);
        assert(combinations(n, 1) == n % modulus);
        assert(combinations(n, n) == 1 % modulus);
        assert(combinations(n, 123456789) ==
               combinations(n, n - 123456789));
    }
}

}  // namespace

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_against_pascal_triangle();
    test_large_arguments();

    int query_count;
    uint32_t modulus;
    fast_input >> query_count >> modulus;
    m1une::math::BinomialCoefficientMod combinations(modulus);
    while (query_count--) {
        uint64_t n, k;
        fast_input >> n >> k;
        fast_output << combinations.binom(n, k) << '\n';
    }
}

#line 1 "verify/math/binomial_coefficient_mod.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/binomial_coefficient"

#include <cassert>
#include <cstdint>
#line 1 "utilities/fast_io.hpp"



#include <algorithm>
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#include <type_traits>
#include <utility>
#include <unistd.h>
#include <vector>

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;

// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};

template <class T>
struct is_range<T, std::void_t<
    decltype(std::begin(std::declval<T&>())),
    decltype(std::end(std::declval<T&>()))
>> : std::true_type {};

template <class T>
inline constexpr bool is_range_v = is_range<T>::value;

template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));

template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;

template <class T, class = void>
struct range_stored_value {
    using type = range_value_t<T>;
};

template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
    using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};

template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;

// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};

template <class T, std::size_t N>
struct is_char_array<T[N]>
    : std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};

template <class T>
struct is_string_like
    : std::bool_constant<
          std::is_same_v<std::decay_t<T>, std::string>
          || std::is_same_v<std::decay_t<T>, const char*>
          || std::is_same_v<std::decay_t<T>, char*>
          || is_char_array<std::remove_reference_t<T>>::value
      > {};

template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;

// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};

template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
    : std::true_type {};

template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;

template <class T, class = void>
struct has_static_mod_raw : std::false_type {};

template <class T>
struct has_static_mod_raw<
    T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
    : std::true_type {};

template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;

// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
    std::is_integral_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>
    || std::is_same_v<std::remove_cv_t<T>, __uint128_t>;

template <class T>
inline constexpr bool is_signed_v =
    std::is_signed_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>;

template <class T>
struct make_unsigned {
    using type = std::make_unsigned_t<T>;
};

template <>
struct make_unsigned<__int128_t> {
    using type = __uint128_t;
};

template <>
struct make_unsigned<__uint128_t> {
    using type = __uint128_t;
};

template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;

}  // namespace internal

struct FastInput {
    static constexpr int buffer_size = 1 << 20;

   private:
    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _length;
    int _file_descriptor;
    bool _streaming;

    bool refill() {
        _position = 0;
        if (_streaming) {
            ssize_t length;
            do {
                length = ::read(_file_descriptor, _buffer, buffer_size);
            } while (length < 0 && errno == EINTR);
            if (length <= 0) {
                _length = 0;
                return false;
            }
            _length = int(length);
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_stream(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = read_char_raw();
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = negative ? result * 10 - (c - '0')
                                  : result * 10 + (c - '0');
                c = read_char_raw();
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = result * 10 + T(c - '0');
                c = read_char_raw();
            }
            value = negative ? T(0) - result : result;
        }
        return true;
    }

    bool prepare_number() {
        if (_length - _position >= 64) return true;
        const int remaining = _length - _position;
        if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
        const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
        _position = 0;
        _length = remaining + added;
        if (_length < buffer_size) _buffer[_length] = '\0';
        return _length != 0;
    }

   public:
    explicit FastInput(std::FILE* stream = stdin)
        : _stream(stream),
          _position(0),
          _length(0),
          _file_descriptor(::fileno(stream)),
          _streaming([&] {
              struct stat status;
              return _file_descriptor >= 0
                     && ::fstat(_file_descriptor, &status) == 0
                     && !S_ISREG(status.st_mode);
          }()) {}

    FastInput(const FastInput&) = delete;
    FastInput& operator=(const FastInput&) = delete;

    int read_char_raw() {
        if (_position == _length && !refill()) return EOF;
        return _buffer[_position++];
    }

    bool skip_spaces() {
        int c = read_char_raw();
        while (c != EOF && c <= ' ') c = read_char_raw();
        if (c == EOF) return false;
        --_position;
        return true;
    }

    bool read(char& value) {
        if (!skip_spaces()) return false;
        value = char(read_char_raw());
        return true;
    }

    bool read(std::string& value) {
        if (!skip_spaces()) return false;
        value.clear();
        while (true) {
            const int begin = _position;
            while (_position < _length &&
                   static_cast<unsigned char>(_buffer[_position]) > ' ') {
                ++_position;
            }
            value.append(_buffer + begin, _position - begin);
            if (_position < _length) {
                ++_position;
                return true;
            }
            if (!refill()) return true;
        }
    }

    bool read(bool& value) {
        int x;
        if (!read(x)) return false;
        value = x != 0;
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>,
        bool
    >
    read(T& value) {
        if (_streaming) return read_integer_from_stream(value);
        if (!prepare_number()) return false;
        int c = static_cast<unsigned char>(_buffer[_position++]);
        while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = static_cast<unsigned char>(_buffer[_position++]);
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const int first = c - '0';
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = negative ? result * 100 - (first * 10 + second)
                                      : result * 100 + (first * 10 + second);
                    ++_position;
                } else {
                    result = negative ? result * 10 - first : result * 10 + first;
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const unsigned first = unsigned(c - '0');
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = result * 100 + T(first * 10 + unsigned(second));
                    ++_position;
                } else {
                    result = result * 10 + T(first);
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = negative ? T(0) - result : result;
        }
        if (_position > _length) _position = _length;
        return true;
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>, bool>
    read(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();
        bool negative = false;
        if (c == '-' || c == '+') {
            negative = c == '-';
            c = read_char_raw();
        }

        long double result = 0;
        while ('0' <= c && c <= '9') {
            result = result * 10 + (c - '0');
            c = read_char_raw();
        }
        if (c == '.') {
            long double place = 0.1L;
            c = read_char_raw();
            while ('0' <= c && c <= '9') {
                result += (c - '0') * place;
                place *= 0.1L;
                c = read_char_raw();
            }
        }
        if (c == 'e' || c == 'E') {
            c = read_char_raw();
            bool exponent_negative = false;
            if (c == '-' || c == '+') {
                exponent_negative = c == '-';
                c = read_char_raw();
            }
            int exponent = 0;
            while ('0' <= c && c <= '9') {
                exponent = exponent * 10 + (c - '0');
                c = read_char_raw();
            }
            long double scale = 1;
            long double power = 10;
            while (exponent > 0) {
                if (exponent & 1) scale *= power;
                power *= power;
                exponent >>= 1;
            }
            result = exponent_negative ? result / scale : result * scale;
        }
        value = static_cast<T>(negative ? -result : result);
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>,
        bool
    >
    read(T& value) {
        long long x;
        if (!read(x)) return false;
        if constexpr (internal::has_static_mod_raw_v<T>) {
            if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
                value = T::raw(uint32_t(x));
            } else {
                value = T(x);
            }
        } else {
            value = T(x);
        }
        return true;
    }

    template <class First, class Second>
    bool read(std::pair<First, Second>& value) {
        if (!read(value.first)) return false;
        return read(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>,
        bool
    >
    read(Range& range) {
        using StoredValue = internal::range_stored_value_t<Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        for (auto&& value : range) {
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                bool x;
                if (!read(x)) return false;
                value = x;
            } else {
                if (!read(value)) return false;
            }
        }
        return true;
    }

    template <class First, class Second, class... Rest>
    bool read(First& first, Second& second, Rest&... rest) {
        if (!read(first)) return false;
        return read(second, rest...);
    }

    template <class T>
    FastInput& operator>>(T& value) {
        if (!read(value)) std::abort();
        return *this;
    }
};

struct FastOutput {
    static constexpr int buffer_size = 1 << 20;

   private:
    inline static const auto digit_quads = [] {
        std::array<char, 40000> result{};
        for (int i = 0; i < 10000; i++) {
            int value = i;
            for (int j = 3; j >= 0; j--) {
                result[4 * i + j] = char('0' + value % 10);
                value /= 10;
            }
        }
        return result;
    }();

    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _precision;
    std::chars_format _float_format;
    char _range_separator;
    std::string* _capture = nullptr;

    template <class T>
    std::string format_cell(const T& value) {
        std::string result;
        struct CaptureGuard {
            std::string*& target;
            std::string* previous;
            ~CaptureGuard() { target = previous; }
        } guard{_capture, _capture};
        _capture = &result;
        write(value);
        return result;
    }

    template <class Matrix>
    void write_aligned_matrix(const Matrix& matrix) {
        std::vector<std::vector<std::string>> rows;
        std::vector<std::size_t> widths;
        for (const auto& row : matrix) {
            auto& cells = rows.emplace_back();
            std::size_t column = 0;
            for (const auto& value : row) {
                cells.push_back(format_cell(value));
                if (column == widths.size()) widths.push_back(0);
                widths[column] = std::max(widths[column], cells.back().size());
                ++column;
            }
        }
        bool first = true;
        for (const auto& row : rows) {
            if (!first) write_char('\n');
            first = false;
            for (std::size_t column = 0; column < row.size(); ++column) {
                if (column != 0) write_char(_range_separator);
                for (std::size_t padding = row[column].size();
                     padding < widths[column]; ++padding) {
                    write_char(' ');
                }
                write(row[column]);
            }
        }
    }

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {
        if (_stream == stdout
            && internal::standard_output_instance == nullptr) {
            internal::standard_output_instance = this;
        }
    }

    FastOutput(const FastOutput&) = delete;
    FastOutput& operator=(const FastOutput&) = delete;

    ~FastOutput() {
        flush();
        if (internal::standard_output_instance == this) {
            internal::standard_output_instance = nullptr;
        }
    }

    void flush() {
        if (_position != 0) {
            std::fwrite(_buffer, 1, _position, _stream);
            _position = 0;
        }
        std::fflush(_stream);
    }

    void write_char(char c) {
        if (_capture != nullptr) {
            _capture->push_back(c);
            return;
        }
        if (_position == buffer_size) flush();
        _buffer[_position++] = c;
    }

    void write(const char* s) {
        while (*s != '\0') write_char(*s++);
    }

    void write(const std::string& s) {
        if (_capture != nullptr) {
            _capture->append(s);
            return;
        }
        std::size_t position = 0;
        while (position < s.size()) {
            if (_position == buffer_size) flush();
            const std::size_t copied =
                std::min<std::size_t>(buffer_size - _position, s.size() - position);
            std::memcpy(_buffer + _position, s.data() + position, copied);
            _position += int(copied);
            position += copied;
        }
    }

    void write(char c) {
        write_char(c);
    }

    void write(bool value) {
        write_char(value ? '1' : '0');
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>>
    write(T value) {
        char digits[128];
        auto [end, error] = std::to_chars(
            digits,
            digits + sizeof(digits),
            value,
            _float_format,
            _precision
        );
        if (error != std::errc()) std::abort();
        for (const char* pointer = digits; pointer != end; pointer++) {
            write_char(*pointer);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>
    >
    write(T value) {
        using Raw = std::remove_cv_t<T>;
        using Unsigned = internal::make_unsigned_t<Raw>;

        Unsigned magnitude;
        if constexpr (internal::is_signed_v<Raw>) {
            if (value < 0) {
                write_char('-');
                magnitude = Unsigned(0) - Unsigned(value);
            } else {
                magnitude = Unsigned(value);
            }
        } else {
            magnitude = value;
        }

        if (magnitude == 0) {
            write_char('0');
            return;
        }

        unsigned chunks[16];
        int count = 0;
        while (magnitude >= 10000) {
            const Unsigned quotient = magnitude / 10000;
            chunks[count++] = unsigned(magnitude - quotient * 10000);
            magnitude = quotient;
        }
        if (_capture == nullptr && _position > buffer_size - 64) flush();
        char captured[64];
        char* const begin = _capture != nullptr ? captured : _buffer + _position;
        char* destination = begin;
        const unsigned leading = unsigned(magnitude);
        const char* first = digit_quads.data() + 4 * leading;
        int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
        for (; skip < 4; skip++) *destination++ = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(destination, digits, 4);
            destination += 4;
        }
        if (_capture != nullptr) {
            _capture->append(begin, destination - begin);
        } else {
            _position += int(destination - begin);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>
    >
    write(const T& value) {
        write(value.val());
    }

    template <class First, class Second>
    void write(const std::pair<First, Second>& value) {
        write(value.first);
        write_char(' ');
        write(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>
    >
    write(const Range& range) {
        using StoredValue = internal::range_stored_value_t<const Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        bool first = true;
        for (const auto& value : range) {
            if (!first) write_char(nested ? '\n' : _range_separator);
            first = false;
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                write(static_cast<bool>(value));
            } else {
                write(value);
            }
        }
    }

    template <class First, class... Rest>
    void print(const First& first, const Rest&... rest) {
        write(first);
        ((write_char(' '), write(rest)), ...);
    }

    void println() {
        write_char('\n');
    }

    void set_precision(int precision) {
        _precision = precision;
    }

    void set_fixed(int precision = 6) {
        _float_format = std::chars_format::fixed;
        _precision = precision;
    }

    void set_general(int precision = 6) {
        _float_format = std::chars_format::general;
        _precision = precision;
    }

    void set_range_separator(char separator) {
        _range_separator = separator;
    }

    template <class Matrix>
    void write_aligned(const Matrix& matrix) {
        using Row = internal::range_stored_value_t<const Matrix>;
        using Cell = internal::range_stored_value_t<const Row>;
        static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
                      "write_aligned requires a two-dimensional range");
        static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
                      "write_aligned requires scalar cells");
        write_aligned_matrix(matrix);
    }

    template <class Matrix>
    void println_aligned(const Matrix& matrix) {
        write_aligned(matrix);
        write_char('\n');
    }

    template <class... Args>
    void println(const Args&... args) {
        print(args...);
        write_char('\n');
    }

    template <class T>
    FastOutput& operator<<(const T& value) {
        write(value);
        return *this;
    }
};

}  // namespace utilities
}  // namespace m1une


#line 7 "verify/math/binomial_coefficient_mod.test.cpp"

#line 1 "math/binomial_coefficient_mod.hpp"



#line 8 "math/binomial_coefficient_mod.hpp"

#line 1 "math/number_theory.hpp"



#line 6 "math/number_theory.hpp"
#include <limits>
#include <tuple>
#line 10 "math/number_theory.hpp"

namespace m1une {
namespace math {

namespace internal {

inline long long safe_mod(long long x, long long mod) {
    x %= mod;
    if (x < 0) x += mod;
    return x;
}

inline unsigned __int128 floor_sum_unsigned(unsigned long long n, unsigned long long mod, unsigned long long a,
                                            unsigned long long b) {
    unsigned __int128 answer = 0;
    while (true) {
        if (a >= mod) {
            answer += static_cast<unsigned __int128>(n) * (n - 1) / 2 * (a / mod);
            a %= mod;
        }
        if (b >= mod) {
            answer += static_cast<unsigned __int128>(n) * (b / mod);
            b %= mod;
        }

        const unsigned __int128 y_max = static_cast<unsigned __int128>(a) * n + b;
        if (y_max < mod) break;
        n = static_cast<unsigned long long>(y_max / mod);
        b = static_cast<unsigned long long>(y_max % mod);
        unsigned long long tmp = mod;
        mod = a;
        a = tmp;
    }
    return answer;
}

}  // namespace internal

// Returns (g, x, y), where g = gcd(a, b) is nonnegative and
// a * x + b * y = g. Returns (0, 0, 0) when a = b = 0.
inline std::tuple<long long, long long, long long> extended_gcd(long long a,
                                                               long long b) {
    using i128 = __int128;
    if (a == 0 && b == 0) return {0, 0, 0};

    i128 old_remainder = a;
    i128 remainder = b;
    if (old_remainder < 0) old_remainder = -old_remainder;
    if (remainder < 0) remainder = -remainder;
    i128 old_x = 1;
    i128 x = 0;
    i128 old_y = 0;
    i128 y = 1;

    while (remainder != 0) {
        i128 quotient = old_remainder / remainder;

        i128 next = old_remainder - quotient * remainder;
        old_remainder = remainder;
        remainder = next;

        next = old_x - quotient * x;
        old_x = x;
        x = next;

        next = old_y - quotient * y;
        old_y = y;
        y = next;
    }

    if (a < 0) old_x = -old_x;
    if (b < 0) old_y = -old_y;

#ifndef NDEBUG
    const i128 minimum = std::numeric_limits<long long>::min();
    const i128 maximum = std::numeric_limits<long long>::max();
    assert(old_remainder <= maximum);
    assert(minimum <= old_x && old_x <= maximum);
    assert(minimum <= old_y && old_y <= maximum);
#endif
    return {static_cast<long long>(old_remainder), static_cast<long long>(old_x),
            static_cast<long long>(old_y)};
}

inline long long pow_mod(long long x, unsigned long long exponent, long long mod) {
    assert(mod >= 1);
    if (mod == 1) return 0;

    unsigned long long base = static_cast<unsigned long long>(internal::safe_mod(x, mod));
    unsigned long long result = 1;
    const unsigned long long unsigned_mod = static_cast<unsigned long long>(mod);
    while (exponent > 0) {
        if (exponent & 1) {
            result = static_cast<unsigned long long>(static_cast<unsigned __int128>(result) * base % unsigned_mod);
        }
        base = static_cast<unsigned long long>(static_cast<unsigned __int128>(base) * base % unsigned_mod);
        exponent >>= 1;
    }
    return static_cast<long long>(result);
}

// Returns gcd(a, mod) and x such that a * x is congruent to gcd(a, mod)
// modulo mod. The returned x is in [0, mod / gcd(a, mod)).
inline std::pair<long long, long long> inv_gcd(long long a, long long mod) {
    assert(mod >= 1);
    a = internal::safe_mod(a, mod);
    if (a == 0) return {mod, 0};

    long long s = mod;
    long long t = a;
    long long m0 = 0;
    long long m1 = 1;
    while (t > 0) {
        const long long quotient = s / t;
        s -= t * quotient;
        m0 -= m1 * quotient;

        long long tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    if (m0 < 0) m0 += mod / s;
    return {s, m0};
}

inline long long inv_mod(long long x, long long mod) {
    const auto result = inv_gcd(x, mod);
    assert(result.first == 1);
    return result.second;
}

// Returns the smallest nonnegative solution and the least common multiple of
// the moduli. Returns {0, 0} when the system is inconsistent.
inline std::pair<long long, long long> crt(const std::vector<long long>& remainders,
                                           const std::vector<long long>& moduli) {
    assert(remainders.size() == moduli.size());

    long long r0 = 0;
    long long m0 = 1;
    for (int i = 0; i < int(remainders.size()); i++) {
        assert(moduli[i] >= 1);
        long long r1 = internal::safe_mod(remainders[i], moduli[i]);
        long long m1 = moduli[i];

        if (m0 < m1) {
            long long tmp = r0;
            r0 = r1;
            r1 = tmp;
            tmp = m0;
            m0 = m1;
            m1 = tmp;
        }
        if (m0 % m1 == 0) {
            if (r0 % m1 != r1) return {0, 0};
            continue;
        }

        const auto inverse = inv_gcd(m0, m1);
        const long long gcd = inverse.first;
        const long long reduced_modulus = m1 / gcd;
        const __int128 difference = static_cast<__int128>(r1) - r0;
        if (difference % gcd != 0) return {0, 0};

        __int128 multiplier = difference / gcd % reduced_modulus;
        multiplier = multiplier * inverse.second % reduced_modulus;
        if (multiplier < 0) multiplier += reduced_modulus;

        const __int128 new_modulus = static_cast<__int128>(m0) * reduced_modulus;
        assert(new_modulus <= std::numeric_limits<long long>::max());
        __int128 new_remainder = static_cast<__int128>(r0) + multiplier * m0;
        new_remainder %= new_modulus;
        if (new_remainder < 0) new_remainder += new_modulus;
        r0 = static_cast<long long>(new_remainder);
        m0 = static_cast<long long>(new_modulus);
    }
    return {r0, m0};
}

// Returns sum_{i=0}^{n-1} floor((a * i + b) / mod).
inline long long floor_sum(long long n, long long mod, long long a, long long b) {
    assert(n >= 0);
    assert(mod >= 1);

    const long long normalized_a = internal::safe_mod(a, mod);
    const long long normalized_b = internal::safe_mod(b, mod);
    __int128 answer = (static_cast<__int128>(a) - normalized_a) / mod * n * (n - 1) / 2;
    answer += (static_cast<__int128>(b) - normalized_b) / mod * n;
    answer += internal::floor_sum_unsigned(static_cast<unsigned long long>(n), static_cast<unsigned long long>(mod),
                                           static_cast<unsigned long long>(normalized_a),
                                           static_cast<unsigned long long>(normalized_b));

    assert(answer >= std::numeric_limits<long long>::min());
    assert(answer <= std::numeric_limits<long long>::max());
    return static_cast<long long>(answer);
}

}  // namespace math
}  // namespace m1une


#line 10 "math/binomial_coefficient_mod.hpp"

namespace m1une {
namespace math {

// Binomial coefficients modulo a fixed, not necessarily prime, modulus.
class BinomialCoefficientMod {
   private:
    struct PrimePower {
        uint32_t prime;
        int exponent;
        uint32_t modulus;
        uint32_t crt_multiplier;
        std::vector<uint32_t> unit_factorial_prefix;

        uint32_t multiply(uint32_t lhs, uint32_t rhs) const {
            return uint32_t(uint64_t(lhs) * rhs % modulus);
        }

        uint32_t power(uint32_t base, uint64_t exponent_) const {
            uint32_t result = 1 % modulus;
            while (exponent_ > 0) {
                if (exponent_ & 1) result = multiply(result, base);
                base = multiply(base, base);
                exponent_ >>= 1;
            }
            return result;
        }

        uint64_t factorial_valuation(uint64_t n) const {
            uint64_t result = 0;
            while (n > 0) {
                n /= prime;
                result += n;
            }
            return result;
        }

        uint32_t unit_factorial(uint64_t n) const {
            if (n == 0) return 1 % modulus;
            const uint32_t block_product = unit_factorial_prefix.back();
            uint32_t result = power(block_product, n / modulus);
            result = multiply(result, unit_factorial_prefix[std::size_t(n % modulus)]);
            return multiply(result, unit_factorial(n / prime));
        }

        uint32_t binom(uint64_t n, uint64_t k) const {
            if (k > n) return 0;
            const uint64_t valuation = factorial_valuation(n) - factorial_valuation(k) -
                                       factorial_valuation(n - k);
            if (valuation >= uint64_t(exponent)) return 0;

            const uint32_t numerator = unit_factorial(n);
            const uint32_t denominator =
                multiply(unit_factorial(k), unit_factorial(n - k));
            const uint32_t inverse_denominator =
                uint32_t(inv_mod(denominator, modulus));
            uint32_t result = multiply(numerator, inverse_denominator);
            result = multiply(result, power(prime, valuation));
            return result;
        }
    };

    uint32_t _modulus;
    std::vector<PrimePower> _prime_powers;

   public:
    explicit BinomialCoefficientMod(uint32_t modulus) : _modulus(modulus) {
        assert(modulus >= 1);
        uint32_t remaining = modulus;
        for (uint32_t prime = 2; uint64_t(prime) * prime <= remaining; prime++) {
            if (remaining % prime != 0) continue;
            int exponent = 0;
            uint32_t prime_power = 1;
            do {
                remaining /= prime;
                prime_power *= prime;
                exponent++;
            } while (remaining % prime == 0);
            _prime_powers.push_back(
                PrimePower{prime, exponent, prime_power, 0, {}});
        }
        if (remaining > 1) {
            _prime_powers.push_back(PrimePower{remaining, 1, remaining, 0, {}});
        }

        for (PrimePower& component : _prime_powers) {
            component.unit_factorial_prefix.resize(std::size_t(component.modulus));
            component.unit_factorial_prefix[0] = 1;
            for (uint32_t value = 1; value < component.modulus; value++) {
                component.unit_factorial_prefix[value] =
                    component.unit_factorial_prefix[value - 1];
                if (value % component.prime != 0) {
                    component.unit_factorial_prefix[value] = component.multiply(
                        component.unit_factorial_prefix[value], value);
                }
            }

            const uint32_t other = modulus / component.modulus;
            const uint32_t inverse =
                uint32_t(inv_mod(other, component.modulus));
            component.crt_multiplier =
                uint32_t(uint64_t(other) * inverse % modulus);
        }
    }

    uint32_t modulus() const {
        return _modulus;
    }

    uint32_t binom(uint64_t n, uint64_t k) const {
        if (k > n || _modulus == 1) return 0;
        uint64_t result = 0;
        for (const PrimePower& component : _prime_powers) {
            const uint32_t residue = component.binom(n, k);
            result += uint64_t(residue) * component.crt_multiplier % _modulus;
            result %= _modulus;
        }
        return uint32_t(result);
    }

    uint32_t operator()(uint64_t n, uint64_t k) const {
        return binom(n, k);
    }
};

using ArbitraryModBinomialCoefficient = BinomialCoefficientMod;

}  // namespace math
}  // namespace m1une


#line 9 "verify/math/binomial_coefficient_mod.test.cpp"

namespace {

void test_against_pascal_triangle() {
    constexpr int maximum = 60;
    for (uint32_t modulus = 1; modulus <= 100; modulus++) {
        m1une::math::BinomialCoefficientMod combinations(modulus);
        assert(combinations.modulus() == modulus);

        std::vector<std::vector<uint32_t>> binom(maximum + 1);
        for (int n = 0; n <= maximum; n++) {
            binom[n].resize(n + 1);
            binom[n][0] = 1 % modulus;
            binom[n][n] = 1 % modulus;
            for (int k = 1; k < n; k++) {
                binom[n][k] = (binom[n - 1][k - 1] + binom[n - 1][k]) % modulus;
            }
            for (int k = 0; k <= n; k++) {
                assert(combinations.binom(n, k) == binom[n][k]);
                assert(combinations(n, k) == binom[n][k]);
            }
            assert(combinations.binom(n, uint64_t(n) + 1) == 0);
        }
    }
}

void test_large_arguments() {
    constexpr uint64_t n = 1000000000000000000ULL;
    for (uint32_t modulus : {1U, 2U, 8U, 9U, 12U, 999983U, 1000000U}) {
        m1une::math::ArbitraryModBinomialCoefficient combinations(modulus);
        assert(combinations(n, 0) == 1 % modulus);
        assert(combinations(n, 1) == n % modulus);
        assert(combinations(n, n) == 1 % modulus);
        assert(combinations(n, 123456789) ==
               combinations(n, n - 123456789));
    }
}

}  // namespace

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_against_pascal_triangle();
    test_large_arguments();

    int query_count;
    uint32_t modulus;
    fast_input >> query_count >> modulus;
    m1une::math::BinomialCoefficientMod combinations(modulus);
    while (query_count--) {
        uint64_t n, k;
        fast_input >> n >> k;
        fast_output << combinations.binom(n, k) << '\n';
    }
}

m1une's library

This documentation is automatically generated by online-judge-tools/verification-helper

verify/math/binomial_coefficient_mod.test.cpp

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