排列组合lucas模板

 //codeforces 559C|51nod1486 Gerald and Giant Chess(组合数学+逆元)

 #include <bits/stdc++.h>
 using namespace std;
 #define LL long long
 typedef pair<int,int> pii;
 const int inf = 0x3f3f3f3f;
 const int N =2e5+;
 #define clc(a,b) memset(a,b,sizeof(a))
 const double eps = 1e-;
 const int MOD = 1e9+;
 void fre() {freopen("in.txt","r",stdin);}
 void freout() {freopen("out.txt","w",stdout);}
 inline int read() {int x=,f=;char ch=getchar();while(ch>''||ch<'') {if(ch=='-') f=-;ch=getchar();}while(ch>=''&&ch<='') {x=x*+ch-'';ch=getchar();}return x*f;}

 struct Point{
     int x,y;
     Point(){}
     Point(int _x,int _y):x(_x),y(_y){}
     bool operator <(const Point &rhs) const{
         if(x==rhs.x) return y<rhs.y;
         return x<rhs.x;
     }
 }p[N];

 int f[N];
 int invv[N];
 int inv(int x){
   int ret=,y=MOD-;
   while(y){
     if(y&)ret=1ll*ret*x%MOD;
     y>>=;x=1ll*x*x%MOD;
   }
   return ret;
 }
 int C(int n,int m){
   if(n<m)return ;
   int ret=1ll*f[n]*invv[m]%MOD;
   ret=1ll*ret*invv[n-m]%MOD;
   return ret;
 }
 int lucas(int n,int m){
    if(m == ) return ;
    return 1ll*C(n % MOD, m % MOD) * lucas(n / MOD, m / MOD) % MOD;
 }

 void init(int n,int m){
   f[]=;
   invv[]=;
   for(int i=;i<=n+m+;++i){
     f[i]=1ll*i*f[i-]%MOD;
     invv[i]=inv(f[i]);
   }
 }

 int sum[N];
 int main(){
     int n,m,q;
     scanf("%d%d%d",&n,&m,&q);
     init(n,m);
     for(int i=;i<=q;i++){
         int x,y;
         x=read(),y=read();
         p[i]=Point(x,y);
     }
     p[++q]=Point(n,m);
     sort(p+,p++q);
     for(int i=;i<=q;i++){
         sum[i]=lucas(p[i].x-+p[i].y-,p[i].x-);
         for(int j=;j<i;j++){
             sum[i]=(sum[i]-1LL*sum[j]*lucas(p[i].x-p[j].x+p[i].y-p[j].y,p[i].x-p[j].x)%MOD+MOD)%MOD;
         }
     }
     printf("%d\n",sum[q]);
     return ;
 }