作者:COMEX黄金2502897957 | 来源:互联网 | 2023-05-16 23:13
Lucas定理A、B是非负整数,p是质数。AB写成p进制:Aa[n]a[n-1]a[0],Bb[n]b[n-1]b[0]。则组合数C(A,B)与C(a[n],b[n])*C(a[n-
Lucas定理
A、B是非负整数,p是质数。A B写成p进制:A=a[n]a[n-1]...a[0],B=b[n]b[n-1]...b[0]。
则组合数C(A,B)与C(a[n],b[n])*C(a[n-1],b[n-1])*...*C(a[0],b[0]) mod p同余
即:Lucas(n,m,p)=C(n%p,m%p)*Lucas(n/p,m/p,p)
//快速幂a^b % k
ll PowerMod(ll a, ll b, ll k) {
ll tmp = a, ret = 1;
while (b) {
if (b & 1) ret = ret * tmp % k;
tmp = tmp * tmp % k;
b >>= 1;
}
return ret;
}
//求C(n, m)%p p最大为10^5 n, m可以很大!
ll Lucas(ll n, ll m, ll p) {
ll ret = 1;
while (n && m) {
ll nn = n%p, mm = m%p;
if (nn //fac[nn]为预处理的 fac[n] = n!%p
ret = ret*fac[nn]*PowerMod(fac[mm]*fac[nn-mm]%p, p-2, p)%p;
n /= p;
m /= p;
}
return ret;
}
//C(n, m) % p
Lucas(n, m, p);
AC代码:
#include
#include
#include
#include
using namespace std;
typedef long long ll;
ll fac[100003];
void init(ll p) {
fac[0] = 1;
for (int i=1; i<=p; i++) fac[i] = fac[i-1]*i%p;
}
ll PowerMod(ll a, ll b, ll k) {
ll tmp = a, ret = 1;
while (b) {
if (b & 1) ret = ret * tmp % k;
tmp = tmp * tmp % k;
b >>= 1;
}
return ret;
}
ll Lucas(ll n, ll m, ll p) {
ll ret = 1;
while (n && m) {
ll nn = n%p, mm = m%p;
if (nn ret = ret*fac[nn]*PowerMod(fac[mm]*fac[nn-mm]%p, p-2, p)%p;
n /= p;
m /= p;
}
return ret;
}
int main() {
int T;
ll n, m, p;
cin >> T;
while (T--) {
cin >> n >> m >> p;
init(p);
cout < }
return 0;
}