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;
}