acmでよく見られる組合せ数の計算方法のまとめについて

LL C(LL a, LL b)
{
    if (a < b) return 0;
    LL ret = 1;
    FE(i, a - b + 1, a)
        ret *= i;
    FE(i, 2, b)
        ret /= i;
    return ret;
}

LL fx[MAXN];

void init()
{
    fx[0] = 1;
    FF(i, 1, MAXN)
        fx[i] = fx[i - 1] * i;
}

LL C(LL a, LL b)
{
    if (a < b) return 0;
    return fx[a] / fx[b] / fx[a - b];
}

const int MAXN1 = 1000;
const int MAXN2 = 1000;
LL f[MAXN1][MAXN2];

void init()
{
    FF(i, 0, MAXN1)
        f[i][0] = 1;
    FF(i, 1, MAXN1)
    {
        FE(j, 1, min(i, MAXN2 - 1))
            f[i][j] = (f[i - 1][j] + f[i - 1][j - 1]) % MOD;
    }
}

map  m;

//     
//k 1 -1
void fun(int n, int k)
{
    for (int i = 2; i <= sqrt(n * 1.0); i++)
    {
        while (n % i == 0)
        {
            n /= i;
            m[i] += k;
        }
    }
    if (n > 1)
    {
        m[n] += k;
    }
}

//       
LL quick_pow(LL a, LL b)
{
    LL ret = 1;
    while (b)
    {
        if (b & 1)
        {
            ret *= a;
            ret %= MOD;
        }
        b >>= 1;
        a *= a;
        a %= MOD;
    }
    return ret;
}

//    
LL C(LL a, LL b)
{
    if (a < b || a < 0 || b < 0)
        return 0;
    m.clear();
    LL ret = 1;
    b = min(a - b, b);
    for (int i = 0; i < b; i++)
    {
        fun(a - i, 1);
    }
    for (int i = b; i >= 1; i--)
    {
        fun(i, -1);
    }

    ///          
    for (__typeof(m.begin()) it = m.begin(); it != m.end(); it++)
    {
        if ((*it).second != 0)
        {
            ret *= quick_pow((*it).first, (*it).second);
            ret %= MOD;
        }
    }
    return ret;
}