codeforces Gym - 100633J Ceizenpok’s formula

拓展Lucas

 #include<cstdio>
 #include<cstdlib>
 #include<algorithm>
 #include<cstring>
 #include<vector>
 #define MAXN 100000+10
 #define ll long long
 #define pb push_back
 #define ft first
 #define sc second
 #define mp make_pair
 using namespace std;
 void extgcd(ll a,ll b,ll &x,ll &y){
     if(!b){x=,y=;}
     else{
         ll xx,yy;
         extgcd(b,a%b,xx,yy);
         x=yy;
         y=xx-a/b*yy;
     }
 }
 ll Inv(ll a,ll b){
     ll x,y;
     extgcd(a,b,x,y);
     x=(x%b+b)%b;
     if(!x)x+=b;
     return x;
 }
 ll Pow(ll a,ll b,ll p){
     ll ret=1LL;
     while(b){
         if(b&){(ret*=a)%=p;}
         (a*=a)%=p;
         b>>=;
     }
     return ret;
 }
 ll fac(ll n,ll pi,ll pk){
     if(!n)return 1LL;
     ll ret=1LL;
     for(ll i=;i<pk;i++){
         if(i%pi)(ret*=i)%=pk;
     }
     ret=Pow(ret,n/pk,pk);
     for(ll i=;i<=(n%pk);i++){
         if(i%pi)(ret*=i)%=pk;
     }
     return ret*fac(n/pi,pi,pk)%pk;
     //每隔pi拿走一个,所以是n/pi,不是n/pk
 }
 ll C(ll n,ll m,ll pi,ll pk){
     ll a=fac(n,pi,pk),b=fac(m,pi,pk),c=fac(n-m,pi,pk);
     ll t=0LL;
     for(ll i=n/pi;i;i/=pi)t+=i;
     for(ll i=m/pi;i;i/=pi)t-=i;
     for(ll i=(n-m)/pi;i;i/=pi)t-=i;
     ll ret=a*Inv(b,pk)*Inv(c,pk)%pk;
     (ret*=Pow(pi,t,pk))%=pk;
     return ret;
 }
 ll n,m,p;
 int main()
 {
     ll ans=0LL;
     scanf("%lld%lld%lld",&n,&m,&p);
     for(ll x=p,i=;i<=p;i++){
         if(x%i==){
             ll pk=;
             while(x%i==){
                 pk*=i;
                 x/=i;
             }
             ll t=C(n,m,i,pk);
             (t*=(p/pk))%=p;
             (t*=Inv(p/pk,pk))%=p;
             (ans+=t)%=p;
         }
     }
     printf("%lld\n",ans);
     return ;
 }