1119 机器人走方格 V2（卢卡斯大组合数取膜）

思路：f(i,j)表示机器人从左上角到(i,j)的方法数，则f(i,j)=f(i-1,j)+f(i,j-1)，这是因为机器人只能走下或走右。模拟一遍可得f(i,j)=C(i+j-2,m-1),由于i,j比较大，所以利用卢卡斯大组合数取膜。

#include <iostream>
#include <queue>
#include <stack>
#include <cstdio>
#include <vector>
#include <map>
#include <set>
#include <bitset>
#include <algorithm>
#include <cmath>
#include <cstring>
#include <cstdlib>
#include <string>
#include <sstream>
#include <time.h>
#define x first
#define y second
#define pb push_back
#define mp make_pair
#define lson l,m,rt*2
#define rson m+1,r,rt*2+1
#define mt(A,B) memset(A,B,sizeof(A))
#define mod 1000000007
using namespace std;
typedef long long LL;
const double PI = acos(-1);
const int N=1e5+10;
const int inf = 0x3f3f3f3f;
const LL INF=0x3f3f3f3f3f3f3f3fLL;
LL exp_mod(LL a, LL b, LL p)
{
    LL res = 1;
    while(b != 0)
    {
        if(b&1) res = (res * a) % p;
        a = (a*a) % p;
        b >>= 1;
    }
    return res;
}
LL Comb(LL a, LL b, LL p)
{
    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(LL i = 0; i < b; ++i)
    {
        ca = (ca * (a - i))%p;
        cb = (cb * (b - i))%p;
    }
    ans = (ca*exp_mod(cb, p - 2, p)) % p;
    return ans;
}
LL Lucas(int n, int m, int p)
{
    LL ans = 1;
    while(n&&m&&ans)
    {
        ans = (ans*Comb(n%p, m%p, p)) % p;
        n /= p;
        m /= p;
    }
    return ans;
}
int main()
{
#ifdef Local
    freopen("data.txt","r",stdin);
#endif
     int m,n;
     cin>>n>>m;
     cout<<Lucas(n+m-2,m-1,mod)<<endl;
#ifdef Local
    cerr << "time: " << (LL) clock() * 1000 / CLOCKS_PER_SEC << " ms" << endl;
#endif
}