数论——lucas定理

网上证明很多，虽然没看懂。。。。

主要解决大组合数取模的情况

费马小定理求大组合数：

a^(p-1)=1%p;

两边同除a

a^(p-2)=1/a%p;

C（n,m）= n!/(m!*(n-m)!)

所以C（n,m）=f(n)*qpow(f(m)*f(n-m),MOD-2))%MOD

预处理组合数f

百度之星2016 1003

先推公式，再lucas

p很大的情况 1e9+7

 #include<iostream>
 #include<string>
 #include<algorithm>
 #include<cstdlib>
 #include<cstdio>
 #include<set>
 #include<map>
 #include<vector>
 #include<cstring>
 #include<stack>
 #include<cmath>
 #include<queue>
 #define clc(a,b) memset(a,b,sizeof(a))
 #include <bits/stdc++.h>
 const int maxn = ;
 const int inf=0x3f3f3f3f;
 const double pi=acos(-);
 typedef long long LL;
 using namespace std;
 const LL MOD = 1e9+;

 LL exp_mod(LL a, LL b, LL p)
 {
     LL res = ;
     while(b != )
     {
         if(b&) res = (res * a) % p;
         a = (a*a) % p;
         b >>= ;
     }
     return res;
 }

 LL Comb(LL a, LL b, LL p)
 {
     if(a < b)   return ;
     if(a == b)  return ;
     if(b > a - b)   b = a - b;

     LL ans = , ca = , cb = ;
     for(LL i = ; i < b; ++i)
     {
         ca = (ca * (a - i))%p;
         cb = (cb * (b - i))%p;
     }
     ans = (ca*exp_mod(cb, p - , p)) % p;
     return ans;
 }

 LL Lucas(int n, int m, int p)
 {
     LL ans = ;

     while(n&&m&&ans)
     {
         ans = (ans*Comb(n%p, m%p, p)) % p;
         n /= p;
         m /= p;
     }
     return ans;
 }

 int main()
 {
     int n, m;
     LL p;
     while(~scanf("%d%d", &n, &m))
     {
         p=MOD;
         printf("%I64d\n", Lucas(n+m-, m-, p));
     }
     return ;
 }

p在100000左右

HDU 3037

 #include<iostream>
 #include<string>
 #include<algorithm>
 #include<cstdlib>
 #include<cstdio>
 #include<set>
 #include<map>
 #include<vector>
 #include<cstring>
 #include<stack>
 #include<cmath>
 #include<queue>
 #define clc(a,b) memset(a,b,sizeof(a))
 #include <bits/stdc++.h>
 const int maxn = ;
 const int inf=0x3f3f3f3f;
 const double pi=acos(-);
 typedef long long LL;
 using namespace std;
 //const LL MOD = 1e9+7;

 LL PowMod(LL a,LL b,LL MOD){
     LL ret=;
     while(b){
         if(b&) ret=(ret*a)%MOD;
         a=(a*a)%MOD;
         b>>=;
     }
     return ret;
 }
 LL fac[];
 LL Get_Fact(LL p){
     fac[]=;
     for(int i=;i<=p;i++)
         fac[i]=(fac[i-]*i)%p;
 }
 LL Lucas(LL n,LL m,LL p){
     LL ret=;
     while(n&&m){
         LL a=n%p,b=m%p;
         if(a<b) return ;
         ret=(ret*fac[a]*PowMod(fac[b]*fac[a-b]%p,p-,p))%p;
         n/=p;
         m/=p;
     }
     return ret;
 }
 int main(){
     int t;
     scanf("%d",&t);
     while(t--){
         LL n,m,p;
         scanf("%I64d%I64d%I64d",&n,&m,&p);
         Get_Fact(p);
         printf("%I64d\n",Lucas(n+m,m,p));
     }
     return ;
 }