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#define PROBLEM "https://judge.yosupo.jp/problem/binomial_coefficient" #include "math/binomial_coefficient.hpp" #include <iostream> int main() { int t, m; std::cin >> t >> m; using binom = binomial_coefficient_arbitrary_mod; binom::set_mod(m); while (t--) { long long n, k; std::cin >> n >> k; std::cout << binom::C(n, k) << '\n'; } }
#line 1 "test/library-checker/binomial_coefficient.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/binomial_coefficient" #line 2 "math/binomial_coefficient.hpp" #include <vector> template <typename mint> struct factorial_table { static mint val(int i) { ensure(i); return facts[i]; } static mint inv(int i) { ensure(i); return ifacts[i]; } static void ensure(int n) { int sz = facts.size(); if (sz > n) return; if (n < sz << 1) n = std::min(mint::mod() - 1, sz << 1); facts.resize(n + 1); ifacts.resize(n + 1); for (int i = sz; i <= n; i++) facts[i] = facts[i - 1] * i; ifacts[n] = facts[n].inv(); for (int i = n; i >= sz; i--) ifacts[i - 1] = ifacts[i] * i; } private: static inline std::vector<mint> facts{1}; static inline std::vector<mint> ifacts{1}; }; template <typename mint> struct binomial_coefficient { using facts = factorial_table<mint>; static mint C(int n, int k) { if (n < 0 || n < k || k < 0) return 0; return facts::val(n) * facts::inv(n - k) * facts::inv(k); } static mint P(int n, int k) { if (n < 0 || n < k || k < 0) return 0; return facts::val(n) * facts::inv(n - k); } static mint H(int n, int k) { if (n < 0 || k < 0) return 0; if (k == 0) return 1; return C(n + k - 1, k); } }; #line 1 "atcoder/math.hpp" #include <algorithm> #include <cassert> #include <tuple> #line 8 "atcoder/math.hpp" #line 1 "atcoder/internal_math.hpp" #include <utility> #ifdef _MSC_VER #include <intrin.h> #endif namespace atcoder { namespace internal { // @param m `1 <= m` // @return x mod m constexpr long long safe_mod(long long x, long long m) { x %= m; if (x < 0) x += m; return x; } // Fast modular multiplication by barrett reduction // Reference: https://en.wikipedia.org/wiki/Barrett_reduction // NOTE: reconsider after Ice Lake struct barrett { unsigned int _m; unsigned long long im; // @param m `1 <= m < 2^31` explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) { } // @return m unsigned int umod() const { return _m; } // @param a `0 <= a < m` // @param b `0 <= b < m` // @return `a * b % m` unsigned int mul(unsigned int a, unsigned int b) const { // [1] m = 1 // a = b = im = 0, so okay // [2] m >= 2 // im = ceil(2^64 / m) // -> im * m = 2^64 + r (0 <= r < m) // let z = a*b = c*m + d (0 <= c, d < m) // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) // < 2^64 * 2 // ((ab * im) >> 64) == c or c + 1 unsigned long long z = a; z *= b; #ifdef _MSC_VER unsigned long long x; _umul128(z, im, &x); #else unsigned long long x = (unsigned long long)(((unsigned __int128)(z)*im) >> 64); #endif unsigned int v = (unsigned int)(z - x * _m); if (_m <= v) v += _m; return v; } }; // @param n `0 <= n` // @param m `1 <= m` // @return `(x ** n) % m` constexpr long long pow_mod_constexpr(long long x, long long n, int m) { if (m == 1) return 0; unsigned int _m = (unsigned int)(m); unsigned long long r = 1; unsigned long long y = safe_mod(x, m); while (n) { if (n & 1) r = (r * y) % _m; y = (y * y) % _m; n >>= 1; } return r; } // Reference: // M. Forisek and J. Jancina, // Fast Primality Testing for Integers That Fit into a Machine Word // @param n `0 <= n` constexpr bool is_prime_constexpr(int n) { if (n <= 1) return false; if (n == 2 || n == 7 || n == 61) return true; if (n % 2 == 0) return false; long long d = n - 1; while (d % 2 == 0) d /= 2; constexpr long long bases[3] = {2, 7, 61}; for (long long a : bases) { long long t = d; long long y = pow_mod_constexpr(a, t, n); while (t != n - 1 && y != 1 && y != n - 1) { y = y * y % n; t <<= 1; } if (y != n - 1 && t % 2 == 0) { return false; } } return true; } template <int n> constexpr bool is_prime = is_prime_constexpr(n); // @param b `1 <= b` // @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) { a = safe_mod(a, b); if (a == 0) return {b, 0}; // Contracts: // [1] s - m0 * a = 0 (mod b) // [2] t - m1 * a = 0 (mod b) // [3] s * |m1| + t * |m0| <= b long long s = b, t = a; long long m0 = 0, m1 = 1; while (t) { long long u = s / t; s -= t * u; m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b // [3]: // (s - t * u) * |m1| + t * |m0 - m1 * u| // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u) // = s * |m1| + t * |m0| <= b auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } // by [3]: |m0| <= b/g // by g != b: |m0| < b/g if (m0 < 0) m0 += b / s; return {s, m0}; } // Compile time primitive root // @param m must be prime // @return primitive root (and minimum in now) constexpr int primitive_root_constexpr(int m) { if (m == 2) return 1; if (m == 167772161) return 3; if (m == 469762049) return 3; if (m == 754974721) return 11; if (m == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; int x = (m - 1) / 2; while (x % 2 == 0) x /= 2; for (int i = 3; (long long)(i)*i <= x; i += 2) { if (x % i == 0) { divs[cnt++] = i; while (x % i == 0) { x /= i; } } } if (x > 1) { divs[cnt++] = x; } for (int g = 2;; g++) { bool ok = true; for (int i = 0; i < cnt; i++) { if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) { ok = false; break; } } if (ok) return g; } } template <int m> constexpr int primitive_root = primitive_root_constexpr(m); // @param n `n < 2^32` // @param m `1 <= m < 2^32` // @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64) unsigned long long floor_sum_unsigned(unsigned long long n, unsigned long long m, unsigned long long a, unsigned long long b) { unsigned long long ans = 0; while (true) { if (a >= m) { ans += n * (n - 1) / 2 * (a / m); a %= m; } if (b >= m) { ans += n * (b / m); b %= m; } unsigned long long y_max = a * n + b; if (y_max < m) break; // y_max < m * (n + 1) // floor(y_max / m) <= n n = (unsigned long long)(y_max / m); b = (unsigned long long)(y_max % m); std::swap(m, a); } return ans; } } // namespace internal } // namespace atcoder #line 10 "atcoder/math.hpp" namespace atcoder { long long pow_mod(long long x, long long n, int m) { assert(0 <= n && 1 <= m); if (m == 1) return 0; internal::barrett bt((unsigned int)(m)); unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m)); while (n) { if (n & 1) r = bt.mul(r, y); y = bt.mul(y, y); n >>= 1; } return r; } long long inv_mod(long long x, long long m) { assert(1 <= m); auto z = internal::inv_gcd(x, m); assert(z.first == 1); return z.second; } // (rem, mod) std::pair<long long, long long> crt(const std::vector<long long>& r, const std::vector<long long>& m) { assert(r.size() == m.size()); int n = int(r.size()); // Contracts: 0 <= r0 < m0 long long r0 = 0, m0 = 1; for (int i = 0; i < n; i++) { assert(1 <= m[i]); long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i]; if (m0 < m1) { std::swap(r0, r1); std::swap(m0, m1); } if (m0 % m1 == 0) { if (r0 % m1 != r1) return {0, 0}; continue; } // assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1) // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1)); // r2 % m0 = r0 // r2 % m1 = r1 // -> (r0 + x*m0) % m1 = r1 // -> x*u0*g = r1-r0 (mod u1*g) (u0*g = m0, u1*g = m1) // -> x = (r1 - r0) / g * inv(u0) (mod u1) // im = inv(u0) (mod u1) (0 <= im < u1) long long g, im; std::tie(g, im) = internal::inv_gcd(m0, m1); long long u1 = (m1 / g); // |r1 - r0| < (m0 + m1) <= lcm(m0, m1) if ((r1 - r0) % g) return {0, 0}; // u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1) long long x = (r1 - r0) / g % u1 * im % u1; // |r0| + |m0 * x| // < m0 + m0 * (u1 - 1) // = m0 + m0 * m1 / g - m0 // = lcm(m0, m1) r0 += x * m0; m0 *= u1; // -> lcm(m0, m1) if (r0 < 0) r0 += m0; } return {r0, m0}; } long long floor_sum(long long n, long long m, long long a, long long b) { assert(0 <= n && n < (1LL << 32)); assert(1 <= m && m < (1LL << 32)); unsigned long long ans = 0; if (a < 0) { unsigned long long a2 = internal::safe_mod(a, m); ans -= 1ULL * n * (n - 1) / 2 * ((a2 - a) / m); a = a2; } if (b < 0) { unsigned long long b2 = internal::safe_mod(b, m); ans -= 1ULL * n * ((b2 - b) / m); b = b2; } return ans + internal::floor_sum_unsigned(n, m, a, b); } } // namespace atcoder #line 2 "math/factorize.hpp" #include <cmath> #include <numeric> long long modmul(long long x, long long y, long long mod) { using i128 = __int128_t; return (long long)(i128(x) * i128(y) % i128(mod)); } long long modpow(long long a, long long n, long long mod) { long long ret = 1; while (n > 0) { if (n & 1) ret = modmul(ret, a, mod); a = modmul(a, a, mod); n >>= 1; } return ret; } long long rho(long long n) { long long z = 0; auto f = [&](long long x) -> long long { long long ret = modmul(x, x, n) + z; if (ret == n) return 0; return ret; }; while (true) { long long x = ++z; long long y = f(x); while (true) { long long d = std::gcd(std::abs(x - y), n); if (d == n) break; if (d > 1) return d; x = f(x); y = f(f(y)); } } } #include <initializer_list> bool miller_rabin(long long n) { if (n == 1) return 0; long long d = n - 1, s = 0; while (~d & 1) d >>= 1, s++; auto check = [&](long long a) -> bool { long long x = modpow(a, d, n); if (x == 1) return 1; long long y = n - 1; for (int i = 0; i < s; i++) { if (x == y) return true; x = modmul(x, x, n); } return false; }; for (long long a : {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}) { if (a >= n) break; if (!check(a)) return false; } return true; } #include <map> #include <queue> std::map<long long, int> factorize(long long n) { std::map<long long, int> ret; while (~n & 1) n >>= 1, ret[2]++; std::queue<long long> q; q.push(n); while (!q.empty()) { long long p = q.front(); q.pop(); if (p == 1) continue; if (miller_rabin(p)) { ret[p]++; continue; } long long d = rho(p); q.push(d); q.push(p / d); } return ret; } #line 49 "math/binomial_coefficient.hpp" struct binomial_coefficient_arbitrary_mod { static void set_mod(int mod) { assert(1 <= mod); m = mod; factors = factorize(m); f.assign(factors.size(), {}); inv_f.assign(factors.size(), {}); } static long long C(long long n, long long k) { if (m == 1 || n < 0 || n < k || k < 0) return 0; ensure(n); long long r = n - k; std::vector<long long> rems(factors.size()), mods(factors.size()); int id = 0; for (auto [p, q] : factors) { long long p_q = pow_ll(p, q); mods[id] = p_q; long long e1 = 0, e2 = 0; for (long long p_i = p_q;;) { e1 += n / p_i - k / p_i - r / p_i; if (p_i > n / p) break; p_i *= p; } for (long long p_i = p;;) { e2 += n / p_i - k / p_i - r / p_i; if (p_i > n / p) break; p_i *= p; } atcoder::internal::barrett bt((unsigned int)(p_q)); long long delta = p == 2 && q >= 3 ? 1 : -1; long long rem = delta == -1 && e1 & 1 ? p_q - 1 : 1; rem = bt.mul(rem, atcoder::pow_mod(p, e2, p_q)); for (long long p_i = 1;;) { rem = bt.mul(rem, f[id][(n / p_i) % p_q]); rem = bt.mul(rem, inv_f[id][(k / p_i) % p_q]); rem = bt.mul(rem, inv_f[id][(r / p_i) % p_q]); if (p_i > n / p) break; p_i *= p; } rems[id] = rem; id++; } return atcoder::crt(rems, mods).first; } private: static void ensure(long long n) { if (max_size > n) return; int id = 0; for (auto [p, q] : factors) { long long p_q = pow_ll(p, q); int sz = f[id].size(); if ((long long)sz > std::min(p_q - 1, n) + 1) continue; f[id].resize(std::min(p_q - 1, n) + 1); inv_f[id].resize(std::min(p_q - 1, n) + 1); max_size = std::max(max_size, std::min(p_q - 1, n) + 1); atcoder::internal::barrett bt((unsigned int)(p_q)); for (int i = sz; i <= std::min(p_q - 1, n); i++) { if (i == 0) { f[id][i] = 1; } else { if (i % p == 0) { f[id][i] = f[id][i - 1]; } else { f[id][i] = bt.mul(f[id][i - 1], i); } } inv_f[id][i] = atcoder::inv_mod(f[id][i], p_q); } id++; } } static long long pow_ll(long long x, long long n) { assert(0 <= n && 1 <= m); long long r = 1, y = x; while (n) { if (n & 1) r *= y; n >>= 1; if (n) y *= y; } return r; } static inline long long m = -1; static inline long long max_size = 0; static inline std::map<long long, int> factors{}; static inline std::vector<std::vector<long long>> f{}; static inline std::vector<std::vector<long long>> inv_f{}; }; #line 3 "test/library-checker/binomial_coefficient.test.cpp" #include <iostream> int main() { int t, m; std::cin >> t >> m; using binom = binomial_coefficient_arbitrary_mod; binom::set_mod(m); while (t--) { long long n, k; std::cin >> n >> k; std::cout << binom::C(n, k) << '\n'; } }