AGC001 E - BBQ Hard 组合数学

题目链接

AGC001 E - BBQ Hard

题解

考虑\(C(n+m,n)\)的组合意义
从\((0,0)\)走到\((n,m)\)的方案数
从\((x,y)\)走到\((x+n,y+m)\)的方案数
考虑\(C(a_i+b_i+a_j+b_j,a_i+b_i)\)的组合意义
从\((0,0)\)走到\((a_i+a_j,b_i+b_j)\)的方案数
从\((-a_i,-b_i)\)走到\((a_j,b_j)\)的方案数
考虑计算任意\((-a_i,-b_i)\)到任意\((a_i,b_i)\)的方案数
减去从自己到自己的就好了

代码

#include<cstdio> 
#include<algorithm> 
#include<cstring> 
#define gc getchar() 
#define pc putchar 
inline int read() { 
	int x = 0,f = 1; 
	char c = gc; 
	while(c < '0' || c > '9') c = gc; 
	while(c <= '9' && c >= '0') x = x * 10 + c - '0',c = gc; 
	return x * f ;
} 
void print(int x) { 
	if(x >= 10 ) print(x / 10); 
	pc(x % 10 + '0'); 
} 
int n; 
const int mod = 1e9 + 7; 
inline int fstpow(int x,int k ){  
	int ret = 1; 
	for(;k;k >>= 1,x = 1ll * x * x % mod)  
		if(k & 1) ret = 1ll * ret * x % mod; 
	return ret; 
} 

const int maxn = 25001; 
int a[200006],b[200007]; 
int jc[(maxn << 2)],inv[(maxn << 2) + 7]; 
inline int C(int x,int y) { 
	return 1ll * jc[x] * inv[y] % mod * inv[x - y]% mod; 
} 
int main() { 
	n = read(); 
	int ans = 0; 
	for(int i = 1;i <= n;++ i) { 
		a[i] = read(),b[i] = read(); 
	} 
	for(int i = 1;i < (maxn << 1);++ i) 
		for(int j = 1;j <= (maxn << 1);++ j)  
	for(int i = 1;i <= n;++ i) { 
	} 
	
	jc[0] = jc[1] = 1; 
	for(int i = 2;i < (maxn << 2); ++ i) jc[i] = 1ll * jc[i - 1] * i % mod;  
	
	inv[(maxn << 2) - 1] = fstpow(jc[(maxn << 2) - 1],mod - 2); 
	print(fstpow(jc[500000],mod - 2)); 
	for(int i = (maxn << 2) - 2;i;-- i)  inv[i] = 1ll * inv[i + 1] * (i + 1) % mod; 
	for(int i = 1;i <= n;++ i) { 
		ans = ((ans - C(a[i] * 2 + b[i] * 2,a[i] * 2)) % mod + mod) % mod; 
	} 
	ans = (1ll * 500000004 * ans) % mod; 
	print(ans); 
	return 0; 
}