【Lucas定理】组合数取模算法

• Step1 Problem:

原题组合数在取模下的算法，用作模板保存。

• Step2 Ideas:

学习链接 学习链接1

• Step3 code：

#define _CRT_SECURE_NO_WARNINGS
#include<cstdio>
#include<iostream>
#include<cstring>
#include<algorithm>
#include<bitset>
#include<cassert>
#include<cctype>
#include<cmath>
#include<cstdlib>
#include<ctime>
#include<deque>
#include<iomanip>
#include<list>
#include<map>
#include<queue>
#include<set>
#include<stack>
#include<vector>
#define lt k<<1
#define rt k<<1|1
#define lowbit(x) x&(-x)
#define lson l,mid,lt
#define rson mid+1,r,rt
using namespace std;
typedef long long  ll;
typedef long double ld;
#define ios ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#define mem(a, b) memset(a, b, sizeof(a))
//#define int ll
const double pi = acos(-1.0);
const double eps = 1e-6;
const ll mod = 1e9 + 7;
const int inf = 0x3f3f3f3f;
const ll INF = 0x3f3f3f3f3f3f3f3f;
const int maxn = 2e5 + 5;
int T, n, m, p;
ll a[maxn], b[maxn];
ll lucas(int x, int y)
{
    if (x < y) return 0;
    else if (x < p) return b[x] * a[y] * a[x - y] % p;
    else return lucas(x / p, y / p) * lucas(x % p, y % p) % p;
}
int main()
{
#ifdef ONLINE_JUDGE
#else 
    freopen("input.txt", "r", stdin);
#endif
    cin >> T;
    while (T--)
    {
        cin >> n >> m >> p;
        a[0] = a[1] = b[0] = b[1] = 1;
        for (int i = 2; i <= n + m; i++) b[i] = b[i - 1] * i % p;//求阶乘
        for (int i = 2; i <= n + m; i++) a[i] = (p - p / i) * a[p % i] % p;//求逆元
        for (int i = 2; i <= n + m; i++) a[i] = a[i - 1] * a[i] % p;
        cout << lucas(n + m, m) << endl;
    }

    return 0;
}