cp-library-cpp
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test/combinatorics/factorial/2.test.cpp
- View this file on GitHub
- Last update: 2021-10-28 01:58:02+09:00
- Problem: https://atcoder.jp/contests/abc156/tasks/abc156_d
Depends on
- Modint (for compile-time constant modulo)
(include/algebra/static_modint.hpp)
- Factorial, Permutation, Combination, Multinomial coefficient
(include/combinatorics/factorial.hpp)
Code
#define PROBLEM "https://atcoder.jp/contests/abc156/tasks/abc156_d"
#include <iostream>
#include "../../../include/algebra/static_modint.hpp"
#include "../../../include/combinatorics/factorial.hpp"
int main() {
int N, A, B;
std::cin >> N >> A >> B;
using mint = lib::static_modint<1000000007>;
mint res = mint(2).pow(N) - 1;
res -= lib::combination<int, mint>(N, A);
res -= lib::combination<int, mint>(N, B);
std::cout << res << '\n';
}#line 1 "test/combinatorics/factorial/2.test.cpp"
#define PROBLEM "https://atcoder.jp/contests/abc156/tasks/abc156_d"
#include <iostream>
#line 1 "include/algebra/static_modint.hpp"
//! @file static_modint.hpp
#ifndef CP_LIBRARY_STATIC_MODINT_HPP
#define CP_LIBRARY_STATIC_MODINT_HPP
#include <cassert>
#include <cstdint>
#line 10 "include/algebra/static_modint.hpp"
#include <limits>
#include <type_traits>
#define CP_LIBRARY_USE_CONSTEXPR
#ifndef CP_LIBRARY_WARN
# if (CP_LIBRARY_DEBUG_LEVEL >= 1)
//! @brief Print warning message
//! @note You can suppress the warning by uncommenting the following line
# define CP_LIBRARY_WARN(msg) (std::cerr << (msg) << '\n')
// # define CP_LIBRARY_WARN(msg) (static_cast<void>(0))
# else
# define CP_LIBRARY_WARN(msg) (static_cast<void>(0))
# undef CP_LIBRARY_USE_CONSTEXPR
# define CP_LIBRARY_USE_CONSTEXPR constexpr
# endif
# define CP_LIBRARY_WARN_NOT_DEFINED
#endif
#ifndef CP_LIBRARY_ASSERT
//! @brief Assert macro
# define CP_LIBRARY_ASSERT(...) assert(__VA_ARGS__)
# define CP_LIBRARY_ASSERT_NOT_DEFINED
#endif
namespace lib {
//! @brief modint (for compile-time constant modulo)
//! @tparam modulo modulo (e.g. 1000000007).
template <std::int_least32_t modulo,
std::enable_if_t<(1 < modulo) && (modulo < std::numeric_limits<std::int_least32_t>::max() / 2),
std::nullptr_t> = nullptr>
struct static_modint {
private:
std::int_least32_t value;
//! @param n non-zero integer
//! @return multiplicative inverse of n
template <typename Tp>
[[nodiscard]] static std::int_least32_t calc_inverse(Tp n) {
CP_LIBRARY_ASSERT(n != 0);
Tp b = modulo, u = 1, v = 0, t;
while (b > 0) {
t = n / b;
// std::swap is not necessarily constexpr in C++17
// std::swap(n -= t * b, b);
Tp tmp = std::move(n -= t * b);
n = std::move(b);
b = std::move(tmp);
// std::swap(u -= t * v, v);
tmp = std::move(u -= t * v);
u = std::move(v);
v = std::move(tmp);
}
if (u < 0)
u += modulo;
return static_cast<std::int_least32_t>(u);
}
//! @brief Calculate modulo and keep the value within [0, modulo)
//! @param v integer
//! @return integer within [0, modulo)
//! @note Time complexity: O(1)
template <typename Tp>
[[nodiscard]] static constexpr std::int_least32_t clamp_ll(Tp v) noexcept {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wsign-compare"
if (modulo <= v || v < -modulo)
v %= modulo;
#pragma GCC diagnostic pop
if (v < 0)
v += modulo;
return static_cast<std::int_least32_t>(v);
}
//! @brief Calculate modulo and keep the value within [0, modulo)
//! @note Time complexity: O(1)
constexpr void clamp_self() noexcept {
if (0 <= value) {
if (value < modulo)
return;
if (value < modulo * 2)
value -= modulo;
else
value -= modulo * 2;
} else {
if (-modulo < value)
value += modulo;
else if (-modulo * 2 < value)
value += modulo * 2;
else {
value += modulo;
value += modulo * 2;
}
}
}
public:
//! @brief underlying integer type
using type = std::int_least32_t;
//! @return modulo (e.g. 1000000007)
[[nodiscard]] static constexpr type mod() noexcept {
return modulo;
}
//! @brief Create a modint of value 0
constexpr static_modint() noexcept : value(0) {}
//! @brief Create a modint without taking modulo
constexpr static_modint(const type v, bool) noexcept : value(v) {}
//! @brief Create a modint
template <typename ValueType>
constexpr static_modint(const ValueType v) noexcept : value() {
if constexpr (std::is_integral_v<ValueType> && (std::numeric_limits<ValueType>::digits <= 32)) {
value = v;
clamp_self();
} else {
value = clamp_ll(v);
}
}
[[nodiscard]] constexpr static_modint operator+(const static_modint rhs) const noexcept {
return static_modint(value + rhs.value);
}
[[nodiscard]] constexpr static_modint operator-(const static_modint rhs) const noexcept {
return static_modint(value - rhs.value);
}
[[nodiscard]] constexpr static_modint operator*(const static_modint rhs) const noexcept {
return static_modint(static_cast<std::int_least64_t>(value) * rhs.value);
}
[[nodiscard]] static_modint operator/(const static_modint rhs) const {
return static_modint(static_cast<std::int_least64_t>(value) * calc_inverse(rhs.value));
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator%(const static_modint rhs) const {
CP_LIBRARY_WARN("static_modint::operator% : Are you sure you want to do this?");
return static_modint(value % rhs.value);
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator&(const static_modint rhs) const {
CP_LIBRARY_WARN("static_modint::operator& : Are you sure you want to do this?");
return static_modint(value & rhs.value, true);
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator|(const static_modint rhs) const {
CP_LIBRARY_WARN("static_modint::operator| : Are you sure you want to do this?");
return static_modint(value | rhs.value);
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator^(const static_modint rhs) const {
CP_LIBRARY_WARN("static_modint::operator^ : Are you sure you want to do this?");
return static_modint(value ^ rhs.value);
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator<<(const static_modint rhs) const {
CP_LIBRARY_WARN("static_modint::operator<< : Are you sure you want to do this?");
return static_modint(static_cast<std::int_least64_t>(value) << rhs.value);
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator>>(const static_modint rhs) const {
CP_LIBRARY_WARN("static_modint::operator>> : Are you sure you want to do this?");
return static_modint(value >> rhs.value, true);
}
constexpr static_modint& operator+=(const static_modint rhs) noexcept {
value += rhs.value;
if (value >= modulo)
value -= modulo;
return *this;
}
constexpr static_modint& operator-=(const static_modint rhs) noexcept {
value -= rhs.value;
if (value < 0)
value += modulo;
return *this;
}
constexpr static_modint& operator*=(const static_modint rhs) noexcept {
value = clamp_ll(static_cast<std::int_least64_t>(value) * rhs.value);
return *this;
}
static_modint& operator/=(const static_modint rhs) {
value = clamp_ll(static_cast<std::int_least64_t>(value) * calc_inverse(rhs.value));
return *this;
}
CP_LIBRARY_USE_CONSTEXPR static_modint& operator%=(const static_modint rhs) {
CP_LIBRARY_WARN("static_modint::operator%= : Are you sure you want to do this?");
value %= rhs.value;
if (value < 0)
value += modulo;
return *this;
}
CP_LIBRARY_USE_CONSTEXPR static_modint& operator&=(const static_modint rhs) {
CP_LIBRARY_WARN("static_modint::operator&= : Are you sure you want to do this?");
value &= rhs.value;
return *this;
}
CP_LIBRARY_USE_CONSTEXPR static_modint& operator|=(const static_modint rhs) {
CP_LIBRARY_WARN("static_modint::operator|= : Are you sure you want to do this?");
value |= rhs.value;
clamp_self();
return *this;
}
CP_LIBRARY_USE_CONSTEXPR static_modint& operator^=(const static_modint rhs) {
CP_LIBRARY_WARN("static_modint::operator^= : Are you sure you want to do this?");
value ^= rhs.value;
clamp_self();
return *this;
}
CP_LIBRARY_USE_CONSTEXPR static_modint& operator<<=(const static_modint rhs) {
CP_LIBRARY_WARN("operator<<= : Are you sure you want to do this?");
value = clamp_ll(static_cast<std::int_least64_t>(value) << rhs.value);
return *this;
}
CP_LIBRARY_USE_CONSTEXPR static_modint& operator>>=(const static_modint rhs) {
CP_LIBRARY_WARN("operator>>= : Are you sure you want to do this?");
value >>= rhs.value;
return *this;
}
template <typename RhsType>
[[nodiscard]] constexpr static_modint operator+(const RhsType rhs) const noexcept {
return static_modint(value + clamp_ll(rhs));
}
template <typename RhsType>
[[nodiscard]] constexpr static_modint operator-(const RhsType rhs) const noexcept {
return static_modint(value - clamp_ll(rhs));
}
template <typename RhsType>
[[nodiscard]] constexpr static_modint operator*(const RhsType rhs) const noexcept {
return static_modint(static_cast<std::int_least64_t>(value) * clamp_ll(rhs));
}
template <typename RhsType>
[[nodiscard]] static_modint operator/(const RhsType rhs) const {
std::int_least64_t mul = (rhs > 0) ? calc_inverse(rhs) : -calc_inverse(-rhs);
return static_modint(value * mul);
}
template <typename RhsType>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator%(const RhsType rhs) const {
CP_LIBRARY_WARN("static_modint::operator% : Are you sure you want to do this?");
return static_modint(value % rhs, true);
}
template <typename RhsType>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator&(const RhsType rhs) const {
CP_LIBRARY_WARN("static_modint::operator& : Are you sure you want to do this?");
return static_modint(value & rhs, true);
}
template <typename RhsType>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator|(const RhsType rhs) const {
CP_LIBRARY_WARN("static_modint::operator| : Are you sure you want to do this?");
return static_modint(value | rhs);
}
template <typename RhsType>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator^(const RhsType rhs) const {
CP_LIBRARY_WARN("static_modint::operator^ : Are you sure you want to do this?");
return static_modint(value ^ rhs);
}
template <typename RhsType>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator<<(const RhsType rhs) const {
CP_LIBRARY_WARN("static_modint::operator<< : Are you sure you want to do this?");
return static_modint(static_cast<std::int_least64_t>(value) << rhs);
}
template <typename RhsType>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator>>(const RhsType rhs) const {
CP_LIBRARY_WARN("static_modint::operator>> : Are you sure you want to do this?");
return static_modint(value >> rhs, true);
}
template <typename RhsType>
constexpr static_modint& operator+=(const RhsType rhs) noexcept {
value = clamp_ll(static_cast<std::int_least64_t>(value) + rhs);
return *this;
}
template <typename RhsType>
constexpr static_modint& operator-=(const RhsType rhs) noexcept {
value = clamp_ll(static_cast<std::int_least64_t>(value) - rhs);
return *this;
}
template <typename RhsType>
constexpr static_modint& operator*=(const RhsType rhs) noexcept {
value = clamp_ll(static_cast<std::int_least64_t>(value) * clamp_ll(rhs));
return *this;
}
template <typename RhsType>
static_modint& operator/=(const RhsType rhs) {
std::int_least64_t mul = (rhs > 0) ? calc_inverse(rhs) : -calc_inverse(-rhs);
value = clamp_ll(value * mul);
return *this;
}
template <typename RhsType>
CP_LIBRARY_USE_CONSTEXPR static_modint& operator%=(const RhsType rhs) {
CP_LIBRARY_WARN("static_modint::operator%= : Are you sure you want to do this?");
value %= rhs;
return *this;
}
template <typename RhsType>
CP_LIBRARY_USE_CONSTEXPR static_modint& operator&=(const RhsType rhs) {
CP_LIBRARY_WARN("static_modint::operator&= : Are you sure you want to do this?");
value &= rhs;
return *this;
}
template <typename RhsType>
CP_LIBRARY_USE_CONSTEXPR static_modint& operator|=(const RhsType rhs) {
CP_LIBRARY_WARN("static_modint::operator|= : Are you sure you want to do this?");
value |= rhs;
clamp_self();
return *this;
}
template <typename RhsType>
CP_LIBRARY_USE_CONSTEXPR static_modint& operator^=(const RhsType rhs) {
CP_LIBRARY_WARN("static_modint::operator^= : Are you sure you want to do this?");
value ^= rhs;
clamp_self();
return *this;
}
template <typename RhsType>
CP_LIBRARY_USE_CONSTEXPR static_modint& operator<<=(const RhsType rhs) {
CP_LIBRARY_WARN("operator<<= : Are you sure you want to do this?");
value = clamp_ll(static_cast<std::int_least64_t>(value) << rhs);
return *this;
}
template <typename RhsType>
CP_LIBRARY_USE_CONSTEXPR static_modint& operator>>=(const RhsType rhs) {
CP_LIBRARY_WARN("operator>>= : Are you sure you want to do this?");
value >>= rhs;
return *this;
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator!() const {
CP_LIBRARY_WARN("static_modint::operator! : Are you sure you want to do this?");
return value == 0;
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint operator~() const {
CP_LIBRARY_WARN("static_modint::operator~ : Are you sure you want to do this?");
return static_modint(~value);
}
[[nodiscard]] constexpr static_modint operator-() const noexcept {
return static_modint(value == 0 ? 0 : modulo - value, true);
}
[[nodiscard]] constexpr static_modint& operator+() const noexcept {
return *this;
}
constexpr static_modint& operator++() noexcept {
value = ((value + 1 == modulo) ? 0 : value + 1);
return *this;
}
constexpr static_modint& operator--() noexcept {
value = ((value == 0) ? modulo - 1 : value - 1);
return *this;
}
constexpr static_modint operator++(int) noexcept {
std::int_least32_t res = value;
++(*this);
return static_modint(res, true);
}
constexpr static_modint operator--(int) noexcept {
std::int_least32_t res = value;
--(*this);
return static_modint(res, true);
}
[[nodiscard]] constexpr bool operator==(const static_modint rhs) const noexcept {
return value == rhs.value;
}
[[nodiscard]] constexpr bool operator!=(const static_modint rhs) const noexcept {
return value != rhs.value;
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator<(const static_modint rhs) const {
CP_LIBRARY_WARN("static_modint::operator< : Are you sure you want to do this?");
return value < rhs.value;
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator<=(const static_modint rhs) const {
CP_LIBRARY_WARN("static_modint::operator<= : Are you sure you want to do this?");
return value <= rhs.value;
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator>(const static_modint rhs) const {
CP_LIBRARY_WARN("static_modint::operator> : Are you sure you want to do this?");
return value > rhs.value;
}
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator>=(const static_modint rhs) const {
CP_LIBRARY_WARN("static_modint::operator>= : Are you sure you want to do this?");
return value >= rhs.value;
}
template <typename RhsType>
[[nodiscard]] constexpr bool operator==(const RhsType rhs) const noexcept {
return value == rhs;
}
template <typename RhsType>
[[nodiscard]] constexpr bool operator!=(const RhsType rhs) const noexcept {
return value != rhs;
}
template <typename RhsType>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator<(const RhsType rhs) const {
CP_LIBRARY_WARN("static_modint::operator< : Are you sure you want to do this?");
return value < rhs;
}
template <typename RhsType>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator<=(const RhsType rhs) const {
CP_LIBRARY_WARN("static_modint::operator<= : Are you sure you want to do this?");
return value <= rhs;
}
template <typename RhsType>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator>(const RhsType rhs) const {
CP_LIBRARY_WARN("static_modint::operator> : Are you sure you want to do this?");
return value > rhs;
}
template <typename RhsType>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator>=(const RhsType rhs) const {
CP_LIBRARY_WARN("static_modint::operator>= : Are you sure you want to do this?");
return value >= rhs;
}
[[nodiscard]] constexpr operator std::int_least32_t() const {
CP_LIBRARY_WARN("A value of type static_modint has been cast to type std::int_lease32_t.");
return value;
}
//! @brief Read value (64-bit signed integer) from std::istream& is, take modulo, and store it in rhs.
//! @return std::istream& is
friend std::istream& operator>>(std::istream& is, static_modint& rhs) {
std::int_least64_t tmp;
is >> tmp;
if (tmp < -modulo || modulo <= tmp)
tmp %= modulo;
if (tmp < 0)
tmp += modulo;
rhs.value = static_cast<std::int_least32_t>(tmp);
return is;
}
//! @brief Print value to std::ostream& os
//! @return std::ostream& os
friend std::ostream& operator<<(std::ostream& os, static_modint& rhs) {
return os << rhs.value;
}
//! @return multiplicative inverse
[[nodiscard]] static_modint inv() const {
return static_modint(calc_inverse(value), true);
}
//! @tparam index_positive_guaranteed set true if and only if you can promise that index is positive
//! @tparam Tp integer type (deduced from parameter)
//! @param index index. This must be an integer, but doesn't have to be primitive.
//! @return index-th power of the value
//! @note Time complexity: O(log(index))
template <bool index_positive_guaranteed = true, typename Tp = std::int_least32_t>
[[nodiscard]] static_modint pow(Tp index) const noexcept {
if constexpr (!index_positive_guaranteed) {
if (value == 0)
return static_modint(0, true);
if (index == 0)
return static_modint(1, true);
if (index < 0)
return static_modint(value, true).inv().pow<true>(-index);
}
static_modint res(1, true), base(value, true);
while (index > 0) {
if ((index & 1) == 1)
res *= base;
base *= base;
index >>= 1;
}
return res;
}
//! @return a pair (a, b) such that b > 0, value is equal to a * (mult inverse of b), and (a + b) is minimal
[[nodiscard]] constexpr std::pair<std::int_least32_t, std::int_least32_t> to_frac() const noexcept {
std::int_least32_t x = modulo - value, y = value, u = 1, v = 1;
std::pair<std::int_least32_t, std::int_least32_t> res {value, 1};
std::int_least32_t num = value, den = 1;
std::int_least32_t cost = num + den;
while (x > 0) {
if (x <= num) {
std::int_least32_t q = num / x;
num = num % x;
den += q * u;
if (num == 0)
break;
if (num + den < cost) {
cost = num + den;
res.first = num;
res.second = den;
}
}
std::int_least32_t q = y / x;
y = y % x;
v += q * u;
q = x / y;
x = x % y;
u += q * v;
}
return res;
}
};
template <typename LhsType, std::int_least32_t modulo>
[[nodiscard]] constexpr static_modint<modulo> operator+(const LhsType lhs, const static_modint<modulo> rhs) noexcept {
return rhs + lhs;
}
template <typename LhsType, std::int_least32_t modulo>
[[nodiscard]] constexpr static_modint<modulo> operator-(const LhsType lhs, const static_modint<modulo> rhs) noexcept {
return -rhs + lhs;
}
template <typename LhsType, std::int_least32_t modulo>
[[nodiscard]] constexpr static_modint<modulo> operator*(const LhsType lhs, const static_modint<modulo> rhs) noexcept {
return rhs * lhs;
}
template <typename LhsType, std::int_least32_t modulo>
[[nodiscard]] static_modint<modulo> operator/(const LhsType lhs, const static_modint<modulo> rhs) {
return rhs.inv() * lhs;
}
template <typename LhsType, std::int_least32_t modulo>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint<modulo>
operator%(const LhsType lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator% : Are you sure you want to do this?");
return static_modint<modulo>(lhs % static_cast<std::int_least32_t>(rhs), true);
}
template <typename LhsType, std::int_least32_t modulo, std::enable_if_t<std::is_integral_v<LhsType>, std::nullptr_t> = nullptr>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint<modulo>
operator<<(const LhsType lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator<< : Are you sure you want to do this?");
return static_modint<modulo>(static_cast<std::int_least64_t>(lhs) << static_cast<std::int_least32_t>(rhs));
}
template <typename LhsType, std::int_least32_t modulo, std::enable_if_t<std::is_integral_v<LhsType>, std::nullptr_t> = nullptr>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR static_modint<modulo>
operator>>(const LhsType lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator>> : Are you sure you want to do this?");
return static_modint<modulo>(lhs >> static_cast<std::int_least32_t>(rhs));
}
template <typename LhsType, std::int_least32_t modulo>
constexpr LhsType& operator+=(LhsType& lhs, const static_modint<modulo> rhs) noexcept {
return lhs += static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
constexpr LhsType& operator-=(LhsType& lhs, const static_modint<modulo> rhs) noexcept {
return lhs -= static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
constexpr LhsType& operator*=(LhsType& lhs, const static_modint<modulo> rhs) noexcept {
return lhs *= static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
constexpr LhsType& operator/=(LhsType& lhs, const static_modint<modulo> rhs) {
return lhs /= static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
CP_LIBRARY_USE_CONSTEXPR LhsType& operator%=(LhsType& lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator%= : Are you sure you want to do this?");
return lhs %= static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
CP_LIBRARY_USE_CONSTEXPR LhsType& operator&=(LhsType& lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator&= : Are you sure you want to do this?");
return lhs &= static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
CP_LIBRARY_USE_CONSTEXPR LhsType& operator|=(LhsType& lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator|= : Are you sure you want to do this?");
return lhs |= static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
CP_LIBRARY_USE_CONSTEXPR LhsType& operator^=(LhsType& lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator^= : Are you sure you want to do this?");
return lhs ^= static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
CP_LIBRARY_USE_CONSTEXPR LhsType& operator<<=(LhsType& lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("operator<<= : Are you sure you want to do this?");
return lhs <<= static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
CP_LIBRARY_USE_CONSTEXPR LhsType& operator>>=(LhsType& lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("operator>>= : Are you sure you want to do this?");
return lhs >>= static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator<(const LhsType lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator< : Are you sure you want to do this?");
return lhs < static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator<=(const LhsType lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator<= : Are you sure you want to do this?");
return lhs < static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator>(const LhsType lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator> : Are you sure you want to do this?");
return lhs < static_cast<std::int_least32_t>(rhs);
}
template <typename LhsType, std::int_least32_t modulo>
[[nodiscard]] CP_LIBRARY_USE_CONSTEXPR bool operator>=(const LhsType lhs, const static_modint<modulo> rhs) {
CP_LIBRARY_WARN("static_modint::operator>= : Are you sure you want to do this?");
return lhs < static_cast<std::int_least32_t>(rhs);
}
} // namespace lib
#undef CP_LIBRARY_USE_CONSTEXPR
#ifdef CP_LIBRARY_WARN_NOT_DEFINED
# undef CP_LIBRARY_WARN
# undef CP_LIBRARY_WARN_NOT_DEFINED
# ifdef CP_LIBRARY_WARN
# undef CP_LIBRARY_WARN
# endif
#endif
#ifdef CP_LIBRARY_ASSERT_NOT_DEFINED
# undef CP_LIBRARY_ASSERT
# undef CP_LIBRARY_ASSERT_NOT_DEFINED
#endif
#endif // CP_LIBRARY_STATIC_MODINT_HPP
#line 1 "include/combinatorics/factorial.hpp"
//! @file factorial.hpp
//! @brief Factorial, Permutation, Combination, Multinomial coefficient
#ifndef CP_LIBRARY_FACTORIAL_HPP
#define CP_LIBRARY_FACTORIAL_HPP
#include <algorithm>
#include <array>
#line 11 "include/combinatorics/factorial.hpp"
#include <numeric>
#include <type_traits>
#ifndef CP_LIBRARY_WARN
# if (CP_LIBRARY_DEBUG_LEVEL >= 1)
//! @brief Print warning message
//! @note You can suppress the warning by uncommenting the following line
# define CP_LIBRARY_WARN(msg) (std::cerr << (msg) << '\n')
// # define CP_LIBRARY_WARN(msg) (static_cast<void>(0))
# else
# define CP_LIBRARY_WARN(msg) (static_cast<void>(0))
# endif
# define CP_LIBRARY_WARN_NOT_DEFINED
#endif
namespace lib {
namespace internal::factorial_hpp {
template <typename... Ts>
struct is_all_same : std::false_type {};
template <>
struct is_all_same<> : std::true_type {};
template <typename Tp>
struct is_all_same<Tp> : std::true_type {};
template <typename Tp, typename... Ts>
struct is_all_same<Tp, Tp, Ts...> : is_all_same<Tp, Ts...> {};
template <typename... Ts>
[[maybe_unused]] constexpr bool is_all_same_v = is_all_same<Ts...>::value;
template <class Tp, class... Ts>
struct first_type { using type = Tp; };
template <class... Ts>
using first_type_t = typename first_type<Ts...>::type;
} // namespace internal::factorial_hpp
//! @tparam Tp deduced from parameter
//! @tparam ReturnType set appropriately if there is a possibility of overflow (e.g. long long, __int128, modint)
//! @param n non-negative integer (doesn't have to be primitive)
//! @return factorial of n (n!), or number of ways to arrange n distinguishable objects in any order.
//! @note Time complexity: O(n)
template <typename Tp, typename ReturnType = Tp>
[[nodiscard]] ReturnType factorial(const Tp n) {
if (n < 0)
CP_LIBRARY_WARN("n is negative.");
ReturnType res = 1;
for (Tp i = 1; i <= n; ++i)
res *= i;
return res;
}
//! @tparam Tp deduced from parameters
//! @tparam ReturnType set appropriately if there is a possibility of overflow (e.g. long long, __int128, modint)
//! @param n non-negative integer (doesn't have to be primitive)
//! @param r non-negative integer (doesn't have to be primitive)
//! @return nPr, or number of ways to select r out of n distinguishable objects and arrange them in any order.
//! @note Time complexity: O(r)
template <typename Tp, typename ReturnType = Tp>
[[nodiscard]] ReturnType permutation(const Tp n, const Tp r) {
if (n == 0)
CP_LIBRARY_WARN("n is zero.");
if (n < 0) {
CP_LIBRARY_WARN("n is negative.");
return 0;
}
if (r < 0) {
CP_LIBRARY_WARN("r is negative.");
return 0;
}
if (n < r) {
CP_LIBRARY_WARN("n is less than r.");
return 0;
}
ReturnType res = 1;
for (Tp i = n - r + 1; i <= n; ++i)
res *= i;
return res;
}
//! @tparam Tp deduced from parameters
//! @tparam ReturnType set appropriately if there is a possibility of overflow (e.g. long long, __int128, modint)
//! @param n non-negative integer (doesn't have to be primitive)
//! @param r non-negative integer (doesn't have to be primitive)
//! @return nCr, or number of ways to select r out of n distinguishable objects.
//! @note Time complexity: O(r)
template <typename Tp, typename ReturnType = Tp>
[[nodiscard]] ReturnType combination(const Tp n, const Tp r) {
if (n == 0)
CP_LIBRARY_WARN("n is zero.");
if (n < 0) {
CP_LIBRARY_WARN("n is negative.");
return 0;
}
if (r < 0) {
CP_LIBRARY_WARN("r is negative.");
return 0;
}
if (n < r) {
CP_LIBRARY_WARN("n is less than r.");
return 0;
}
ReturnType res = 1;
for (Tp i = 1; i <= r; ++i) {
res *= (n - r + i);
res /= i;
}
return res;
}
//! @tparam Tp deduced from parameters
//! @tparam ReturnType set appropriately if there is a possibility of overflow (e.g. long long, __int128, modint)
//! @tparam Ts deduced from parameters
//! @param n non-negative integer
//! @param r non-negative integers of same type whose sum is less than or equal to n
//! @return Number of ways to arrange n objects when r_1, r_2, ... objects are indistinguishable.
//! @note Time complexity: O(n - max(r))
template <typename Tp, typename ReturnType = Tp, typename... Ts>
[[nodiscard]] ReturnType multinomial(const Tp n, const Ts... r) {
static_assert(internal::factorial_hpp::is_all_same_v<Ts...>);
if (n == 0)
CP_LIBRARY_WARN("n is zero.");
if (n < 0) {
CP_LIBRARY_WARN("n is negative.");
return 0;
}
if (((r < 0) || ...)) {
CP_LIBRARY_WARN("r... contains negative number.");
return 0;
}
if ((r + ...) > n) {
CP_LIBRARY_WARN("Sum of r... is greater than n.");
return 0;
}
std::array<internal::factorial_hpp::first_type_t<Ts...>, sizeof...(Ts)> r_array {r...};
const auto max_idx = std::distance(std::cbegin(r_array), std::max_element(std::cbegin(r_array), std::cend(r_array)));
const auto max_r = r_array[max_idx];
unsigned current_den_idx = static_cast<int>(max_idx == 0);
internal::factorial_hpp::first_type_t<Ts...> current_den_cnt = 1;
ReturnType res = 1;
for (Tp i = 1; i <= n - max_r; ++i) {
res *= (n - i + 1);
res /= current_den_cnt;
if (current_den_idx < sizeof...(Ts) - 1 && (current_den_cnt == r_array[current_den_idx] || current_den_idx == max_idx)) {
current_den_idx += static_cast<int>(current_den_cnt == r_array[current_den_idx]);
current_den_idx += static_cast<int>(current_den_idx == max_idx);
current_den_cnt = 1;
} else if (current_den_idx == sizeof...(Ts) - 1 && current_den_cnt == r_array[current_den_idx]) {
current_den_cnt = 1;
} else {
++current_den_cnt;
}
}
return res;
}
//! @tparam Tp deduced from parameters
//! @tparam ReturnType set appropriately if there is a possibility of overflow (e.g. long long, __int128, modint)
//! @tparam Container container type (deduced from parameters)
//! @param n non-negative integer
//! @param r container of non-negative integers whose sum is less than or equal to n
//! @return Number of ways to arrange n objects when r[0], r[1], ... objects are indistinguishable.
//! @note Time complexity: O(n - max(r))
template <typename Tp, typename ReturnType = Tp, typename Container>
[[nodiscard]] ReturnType multinomial(const Tp n, const Container& r) {
using Elem = std::decay_t<decltype(*std::cbegin(r))>;
if (n == 0)
CP_LIBRARY_WARN("n is zero.");
if (n < 0) {
CP_LIBRARY_WARN("n is negative.");
return 0;
}
if (std::any_of(std::cbegin(r), std::cend(r), [](const auto v) { return v < 0; })) {
CP_LIBRARY_WARN("r contains negative number.");
return 0;
}
if (std::reduce(std::cbegin(r), std::cend(r), Elem(0)) > n) {
CP_LIBRARY_WARN("Sum of r is greater than n.");
return 0;
}
const auto max_idx = std::distance(std::cbegin(r), std::max_element(std::cbegin(r), std::cend(r)));
const auto max_r = r[max_idx];
unsigned current_den_idx = static_cast<int>(max_idx == 0);
Elem current_den_cnt = 1;
ReturnType res = 1;
for (Tp i = 1; i <= n - max_r; ++i) {
res *= (n - i + 1);
res /= current_den_cnt;
if (current_den_idx < std::size(r) - 1 && (current_den_cnt == r[current_den_idx] || current_den_idx == max_idx)) {
current_den_idx += static_cast<int>(current_den_cnt == r[current_den_idx]);
current_den_idx += static_cast<int>(current_den_idx == max_idx);
current_den_cnt = 1;
} else if (current_den_idx == std::size(r) - 1 && current_den_cnt == r[current_den_idx]) {
current_den_cnt = 1;
} else {
++current_den_cnt;
}
}
return res;
}
//! @tparam Tp deduced from parameters
//! @tparam ReturnType set appropriately if there is a possibility of overflow (e.g. long long, __int128, modint)
//! @param n non-negative integer (doesn't have to be primitive)
//! @param r non-negative integer (doesn't have to be primitive)
//! @return nHr, or number of ways to put n indistinguishable balls into r distinguishable bins.
//! @note Time complexity: O(r)
template <typename Tp, typename ReturnType = Tp>
[[nodiscard]] ReturnType stars_and_bars(const Tp n, const Tp r) {
return combination<Tp, ReturnType>(n + r - 1, r);
}
//! @tparam Max upper limit
//! @tparam ReturnType value type
//! @return std::array which contains 0!, 1!, ..., Max! (Max + 1 numbers)
//! @note Time complexity: O(Max)
template <std::size_t Max, typename ReturnType>
[[nodiscard]] std::array<ReturnType, Max + 1> factorial_array() {
std::array<ReturnType, Max + 1> res;
res[0] = 1;
for (std::size_t i = 1; i <= Max; ++i)
res[i] = res[i - 1] * i;
return res;
}
//! @tparam Max upper limit
//! @tparam Modint value type (deduced from parameter, must be Modint)
//! @param fact factorial of Max (which is the result of factorial or factorial_array)
//! @return std::array which contains multiplicative inverse of 0!, 1!, ..., Max! (Max + 1 numbers)
//! @note Time complexity: O(Max)
template <std::size_t Max, typename Modint>
[[nodiscard]] std::array<Modint, Max + 1> factorial_modinv_array(const Modint fact) {
std::array<Modint, Max + 1> res;
res[Max] = fact.inv();
for (std::size_t i = Max; i > 0; --i)
res[i - 1] = res[i] * i;
return res;
}
//! @tparam Size Size of factorial_array and factorial_modinv_array (deduced from parameters)
//! @tparam Tp deduced from parameters
//! @tparam Modint deduced from parameters
//! @param n non-negative integer
//! @param r non-negative integer
//! @param factorial_array Array that factorial_array() returns
//! @param factorial_modinv_array Array that factorial_modinv_array() returns
//! @return nPr, or number of ways to select r out of n distinguishable objects and arrange them in any order.
//! @note Time complexity: O(1)
template <std::size_t Size, typename Modint, typename Tp>
[[nodiscard]] Modint permutation(const Tp n, const Tp r, const std::array<Modint, Size>& factorial_array, const std::array<Modint, Size>& factorial_modinv_array) {
if (n == 0)
CP_LIBRARY_WARN("n is zero.");
if (n < 0) {
CP_LIBRARY_WARN("n is negative.");
return 0;
}
if (r < 0) {
CP_LIBRARY_WARN("r is negative.");
return 0;
}
if (n < r) {
CP_LIBRARY_WARN("n is less than r.");
return 0;
}
return factorial_array[n] * factorial_modinv_array[n - r];
}
//! @tparam Size Size of factorial_array and factorial_modinv_array (deduced from parameters)
//! @tparam Tp deduced from parameters
//! @tparam Modint deduced from parameters
//! @param n non-negative integer
//! @param r non-negative integer
//! @param factorial_array Array that factorial_array() returns
//! @param factorial_modinv_array Array that factorial_modinv_array() returns
//! @return nCr, or number of ways to select r out of n distinguishable objects.
//! @note Time complexity: O(1)
template <std::size_t Size, typename Modint, typename Tp>
[[nodiscard]] Modint combination(const Tp n, const Tp r, const std::array<Modint, Size>& factorial_array, const std::array<Modint, Size>& factorial_modinv_array) {
if (n == 0)
CP_LIBRARY_WARN("n is zero.");
if (n < 0) {
CP_LIBRARY_WARN("n is negative.");
return 0;
}
if (r < 0) {
CP_LIBRARY_WARN("r is negative.");
return 0;
}
if (n < r) {
CP_LIBRARY_WARN("n is less than r.");
return 0;
}
return factorial_array[n] * factorial_modinv_array[n - r] * factorial_modinv_array[r];
}
//! @tparam Size Size of factorial_array and factorial_modinv_array (deduced from parameters)
//! @tparam Modint deduced from parameters
//! @tparam Tp deduced from parameters
//! @tparam Ts deduced from parameters
//! @param n non-negative integer
//! @param r non-negative integers whose sum is less than or equal to n
//! @param factorial_array Array that factorial_array() returns
//! @param factorial_modinv_array Array that factorial_modinv_array() returns
//! @return Number of ways to arrange n objects when r_1, r_2, ... objects are indistinguishable.
//! @note Time complexity: O(sizeof...(r))
template <std::size_t Size, typename Modint, typename Tp, typename... Ts>
[[nodiscard]] Modint multinomial(const Tp n, const Ts... r, const std::array<Modint, Size>& factorial_array, const std::array<Modint, Size>& factorial_modinv_array) {
if (n == 0)
CP_LIBRARY_WARN("n is zero.");
if (n < 0) {
CP_LIBRARY_WARN("n is negative.");
return 0;
}
if (((r < 0) || ...)) {
CP_LIBRARY_WARN("r contains negative number.");
return 0;
}
if ((r + ...) > n) {
CP_LIBRARY_WARN("Sum of r... is greater than n.");
return 0;
}
return factorial_array[n] * ((factorial_modinv_array[r]) * ...);
}
//! @tparam Size Size of factorial_array and factorial_modinv_array (deduced from parameters)
//! @tparam Modint deduced from parameters
//! @tparam Tp deduced from parameters
//! @tparam container type (deduced from parameters)
//! @param n non-negative integer
//! @param r container of non-negative integers whose sum is less than or equal to n
//! @param factorial_array Array that factorial_array() returns
//! @param factorial_modinv_array Array that factorial_modinv_array() returns
//! @return Number of ways to arrange n objects when r[0], r[1], ... objects are indistinguishable.
//! @note Time complexity: O(size(r))
template <std::size_t Size, typename Modint, typename Tp, typename Container>
[[nodiscard]] Modint multinomial(const Tp n, const Container& r, const std::array<Modint, Size>& factorial_array, const std::array<Modint, Size>& factorial_modinv_array) {
using Elem = std::decay_t<decltype(*std::cbegin(r))>;
if (n == 0)
CP_LIBRARY_WARN("n is zero.");
if (n < 0) {
CP_LIBRARY_WARN("n is negative.");
return 0;
}
if (std::any_of(std::cbegin(r), std::cend(r), [](const auto v) { return v < 0; })) {
CP_LIBRARY_WARN("r contains negative number.");
return 0;
}
if (std::reduce(std::cbegin(r), std::cend(r), Elem(0)) > n) {
CP_LIBRARY_WARN("Sum of r is greater than n.");
return 0;
}
return factorial_array[n] * std::reduce(std::cbegin(r), std::cend(r), Modint(1),
[&](const Modint res, const Elem e) { return res * factorial_modinv_array[e]; });
}
//! @tparam Size Size of factorial_array and factorial_modinv_array (deduced from parameters)
//! @tparam Tp deduced from parameters
//! @tparam Modint deduced from parameters
//! @param n non-negative integer
//! @param r non-negative integers whose sum is equal to n
//! @param factorial_array Array that factorial_array() returns
//! @param factorial_modinv_array Array that factorial_modinv_array() returns
//! @return nHr, or number of ways to put n indistinguishable balls into r distinguishable bins.
//! @note Time complexity: O(1)
template <std::size_t Size, typename Modint, typename Tp>
[[nodiscard]] Modint stars_and_bars(const Tp n, const Tp r, const std::array<Modint, Size>& factorial_array, const std::array<Modint, Size>& factorial_modinv_array) {
return combination(n + r - 1, r, factorial_array, factorial_modinv_array);
}
} // namespace lib
#ifdef CP_LIBRARY_WARN_NOT_DEFINED
# undef CP_LIBRARY_WARN
# undef CP_LIBRARY_WARN_NOT_DEFINED
# ifdef CP_LIBRARY_WARN
# undef CP_LIBRARY_WARN
# endif
#endif
#endif
#line 6 "test/combinatorics/factorial/2.test.cpp"
int main() {
int N, A, B;
std::cin >> N >> A >> B;
using mint = lib::static_modint<1000000007>;
mint res = mint(2).pow(N) - 1;
res -= lib::combination<int, mint>(N, A);
res -= lib::combination<int, mint>(N, B);
std::cout << res << '\n';
}