BBQ Hard - AtCoder Grand Contest 001

#include <bits/stdc++.h>
using namespace std;

const int N = 200010;
const int offset = 2010;
const int md = 1e9 + 7;

int n;
int a[N], b[N];
vector<int> _get[offset + offset];
int f[offset + offset][offset + offset];
int upd[offset + offset][offset + offset];
int ans;
int fact[N], inv[N];

void add(int &x, int y) { x += y; x %= md; if(x < 0) x += md; }

int pw(int a, int b) {
    int res = 1;
    for(int i = 0; (1LL << i) <= b; ++i) {
        if (b & (1LL << i)) res = 1LL * res * a % md;
        a = 1LL * a * a % md;
    }
    return res;
}

int main() {
    scanf("%d", &n);
    for(int i = 1; i <= n; ++i) {
        scanf("%d%d", &a[i], &b[i]);
        _get[a[i] + offset].push_back(b[i] + offset);
        upd[-a[i] + offset][-b[i] + offset]++;
    }

    for(int x = 0; x < offset + offset; ++x) {
        for(int y = 0; y < offset + offset; ++y) {
            if(x > 0) add(f[x][y], f[x - 1][y]);
            if(y > 0) add(f[x][y], f[x][y - 1]);
            if(upd[x][y]) add(f[x][y], upd[x][y]);
        }
        for (int y : _get[x]) add(ans, f[x][y]);
    }

    fact[0] = 1; for(int i = 1; i < N; ++i) fact[i] = 1LL * fact[i-1] * i % md;
    for(int i = 0; i < N; ++i) inv[i] = pw(fact[i], md - 2);

    for(int i = 1; i <= n; ++i) {
        int A = a[i] * 2, B = b[i] * 2;
        add(ans, md - (1LL * fact[A + B] * (1LL * inv[A] * inv[B] % md) % md));
    }
    ans = 1LL * ans * pw(2, md - 2) % md;

    printf("%d\n", ans);
}