求大的组合数模板 利用Lucas定理

Posted 2021-01-10 l609929321

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)

ll fact[maxn], a[maxn], inv[maxn]; //fact为阶乘
void init() {
    a[0] = a[1] = 1;
    fact[0] = fact[1] = 1;
    inv[1] = 1;
    for( ll i = 2; i <= 100005; i ++ ) {
        fact[i] = fact[i-1] * i % mod;
        inv[i] = (mod - mod/i)*inv[mod%i]%mod;
        a[i] = a[i-1] * inv[i] % mod;
    }
}

ll C( ll n, ll m ) {
    return fact[n]*a[n-m]%mod*a[m]%mod;
}