Lucas定理是用来求 c(n,m) mod p,p为素数的值。
C(n, m) % p = C(n / p, m / p) * C(n % p,m % p) % p
(Lucas(n,m,p)=C(n \% p,m \% p) imes Lucas(frac{n}{p},frac{m}{p},p))
(inom{n}{m}=frac{n!}{m!(n-m)!})
卢卡斯定理模板
#include <cstdio>
#include <iostream>
#define ll long long
using namespace std;
ll pow(ll a,ll b,ll p){
ll ans = 1; a %= p;
while(b){
if(b & 1)ans = ans * a % p;
b >>= 1;
a = a * a % p;
}
return ans;
}
ll inv(ll x, ll p){//x关于p的逆元,p为素数
return pow(x, p - 2, p);
}
ll C(ll n, ll m, ll p){//求组合数C(n,m)%p
if(n < m) return 0;
ll up = 1, down = 1;//分子分母
for(ll i = n - m + 1; i <= n; i++) up = up * i % p;
for(ll i = 1; i <= m; i++) down = down * i % p;
return up * inv(down, p) % p;
}
ll lucas(ll n, ll m, ll p){
if(m == 0) return 1;
return C(n % p, m % p, p) * lucas(n / p, m / p, p) % p;
}
int main(){
ll n, m, p;
scanf("%lld%lld%lld", &n, &m, &p);
printf("%lld
", lucas(n, m, p));
return 0;
}