Lucas定理求组合数模板

Posted 2020-10-09

$Lucas(n,m,p)=C(n\%p,m\%p)*Lucas(n/p,m/p,p)$

$C^n_m=\frac{n!}{m!(n-m)!}$

$x^{p-1}\equiv 1(mod p)\Longrightarrow x*x^{p-2}\equiv 1(mod p)\Longrightarrow x^{p-2}\equiv (1/x)^p(mod p)$

$C^n_m=n!*[m!(n-m)!]^{p-2}(mod p)$

 1 #include <iostream>
 2 #include <cstdio>
 3 #define ll long long
 4 using namespace std;
 5 ll n,m,p,fac[100001];
 6 ll power(ll a,ll b,ll p)
 7 {
 8     ll t=a%p,s=1;
 9     while(b)
10     {
11         if (b%2==1) s=s*t%p;
12         t=t*t%p;
13         b/=2;
14     }
15     return s;
16 }
17 ll C(ll n,ll m,ll p)
18 {
19     if (m>n) return 0;
20     return fac[n]*power(fac[m]*fac[n-m],p-2,p)%p;
21 }
22 ll Lucas(ll n,ll m,ll p)
23 {
24     if (m==0) return 1;
25     return C(n%p,m%p,p)*Lucas(n/p,m/p,p)%p;
26 }
27 int main()
28 {
29     cin>>n>>m>>p;
30     fac[0]=1;
31     for (ll i=1;i<=p;i++) fac[i]=fac[i-1]*i%p;
32     cout<<Lucas(n,m,p)<<endl;
33 }