[组合数]lucas定理 排列组合取模

lucas： 计算 C 时 当 n， m范围大 mod范围小时可以将 n m 缩减至 mod 范围内计算

预处理： 当 n m 范围小 mod 范围大时可以预处理阶乘逆元 O（1）计算 

const int FN = 1e5 + 10;
const int MOD = 1e9 + 7;
ll fac[FN] = {1, 1}, inv[FN] = {1,1}, f[FN] = {1,1};
ll C(ll a,ll b)
{
	if(b > a) return 0;
	return fac[a] * inv[b] % MOD * inv[a - b] % MOD;
}
void init() //预处理逆元和阶乘
{
	for(int i = 2; i < FN; i++)
	{
		fac[i] = fac[i - 1] * i % MOD;
		f[i] = (MOD - MOD / i) * f[MOD % i] % MOD;
		inv[i] = inv[i - 1] * f[i] % MOD;
	}
}

lucas

 C  m 
    n  = lucas(a, b)

ll mulit(ll a, ll b, ll m)
{
    ll ans = 0;
    while(b)
    {
        if(b & 1)
            ans = (ans + a) % m;
        a = (a << 1) % m;
        b >>= 1;
    }
    return ans;
}
 
ll quick_mod(ll a, ll b, ll m)
{
    ll ans = 1;
    while(b)
    {
        if(b&1)
            ans = mulit(ans, a, m);
        a = mulit(a, a, m);
        b >>= 1;
    }
    return ans;
}
 
ll comp(ll a, ll b, ll m)
{
    if(a < b)
        return 0;
    if(a == b)
        return 1;
    if(b > a - b)
        b = a - b;
    ll ans = 1, ca = 1, cb = 1;
 
    for(int i = 0; i < b; i++)
    {
        ca = ca * (a - i) % m;
        cb = cb * (b - i) % m;
    }
 
    ans = ca * quick_mod(cb, m - 2, m) % m;
    return ans;
}
 
ll lucas(ll a,ll b,ll m)
{
    ll ans = 1;
    while(a && b)
    {
        ans = (ans * comp(a % m, b % m, m)) % m;
        a /= m;
        b /= m;
    }
    return ans;
}