Problem:
Score : 1400 points

Problem Statement

Snuke is having another barbeque party.

This time, he will make one serving of Skewer Meal.

He has a stock of N Skewer Meal Packs. The i-th Skewer Meal Pack contains one skewer, A_i pieces of beef and B_i pieces of green pepper.
All skewers in these packs are different and distinguishable, while all pieces of beef and all pieces of green pepper are, respectively, indistinguishable.

To make a Skewer Meal, he chooses two of his Skewer Meal Packs, and takes out all of the contents from the chosen packs, that is, two skewers and some pieces of beef or green pepper.
(Remaining Skewer Meal Packs will not be used.)
Then, all those pieces of food are threaded onto both skewers, one by one, in any order.

(See the image in the Sample section for better understanding.)

In how many different ways can he make a Skewer Meal? Two ways of making a Skewer Meal is different if and only if the sets of the used skewers are different, or the orders of the pieces of food are different.
Since this number can be extremely large, find it modulo 10^9+7.

Constraints

2≦N≦200,000

1≦A_i≦2000, 1≦B_i≦2000

Input

The input is given from Standard Input in the following format:

N
A_1 B_1
A_2 B_2
:
A_N B_N

Output

Print the number of the different ways Snuke can make a serving of Skewer Meal, modulo 10^9+7.

Sample Input 1

3
1 1
1 1
2 1

Sample Output 1

26

The 26 ways of making a Skewer Meal are shown below.
Gray bars represent skewers, each with a number denoting the Skewer Meal Set that contained the skewer.
Brown and green rectangles represent pieces of beef and green pepper, respectively.

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

// package atcoder.agc.agc001;

import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.InputMismatchException;

/**
 * Created by hama_du on 2016/07/17.
 */
public class Main {
    private static final int MOD = 1000000007;

    public static void main(String[] args) {
        InputReader in = new InputReader(System.in);
        PrintWriter out = new PrintWriter(System.out);
        
        int n = in.nextInt();
        int[][] ab = new int[n][2];
        for (int i = 0; i < n ; i++) {
            for (int j = 0; j < 2 ; j++) {
                ab[i][j] = in.nextInt();
            }
        }
        int GETA = 2005;
        int[][] dp = new int[4010][4010];
        for (int i = 0; i < n ; i++) {
            dp[GETA-ab[i][0]][GETA-ab[i][1]] += 1;
        }
        for (int i = 0 ; i < dp.length-1 ; i++) {
            for (int j = 0 ; j < dp.length-1 ; j++) {
                dp[i+1][j] += dp[i][j];
                if (dp[i+1][j] >= MOD) {
                    dp[i+1][j] -= MOD;
                }
                dp[i][j+1] += dp[i][j];
                if (dp[i][j+1] >= MOD) {
                    dp[i][j+1] -= MOD;
                }
            }
        }

        long sum = 0;
        for (int i = 0; i < n ; i++) {
            sum += dp[GETA+ab[i][0]][GETA+ab[i][1]];
        }
        sum %= MOD;

        prec(10010);

        long dec = 0;
        for (int i = 0; i < n ; i++) {
            dec += comb((ab[i][0]+ab[i][1])*2, ab[i][0]*2);
        }
        dec %= MOD;

        out.println((sum-dec+MOD)*inv(2)%MOD);
        out.flush();
    }

    static long pow(long a, long x) {
        long res = 1;
        while (x > 0) {
            if (x % 2 != 0) {
            res = (res * a) % MOD;
            }
            a = (a * a) % MOD;
            x /= 2;
        }
        return res;
    }

    static long inv(long a) {
        return pow(a, MOD - 2) % MOD;
    }

    static long[] _fact;
    static long[] _invfact;
    static long comb(long ln, long lr) {
        int n = (int)ln;
        int r = (int)lr;
        if (n < 0 || r < 0 || r > n) {
            return 0;
        }
        if (r > n / 2) {
            r = n - r;
        }
        return (((_fact[n] * _invfact[n - r]) % MOD) * _invfact[r]) % MOD;
    }

    static void prec(int n) {
        _fact = new long[n + 1];
        _invfact = new long[n + 1];
        _fact[0] = 1;
        _invfact[0] = 1;
        for (int i = 1; i <= n; i++) {
            _fact[i] = _fact[i - 1] * i % MOD;
            _invfact[i] = inv(_fact[i]);
        }
    }


    static class InputReader {
        private InputStream stream;
        private byte[] buf = new byte[1024];
        private int curChar;
        private int numChars;

        public InputReader(InputStream stream) {
            this.stream = stream;
        }

        private int next() {
            if (numChars == -1)
                throw new InputMismatchException();
            if (curChar >= numChars) {
                curChar = 0;
                try {
                    numChars = stream.read(buf);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
                if (numChars <= 0)
                    return -1;
            }
            return buf[curChar++];
        }

        public char nextChar() {
            int c = next();
            while (isSpaceChar(c))
                c = next();
            if ('a' <= c && c <= 'z') {
                return (char) c;
            }
            if ('A' <= c && c <= 'Z') {
                return (char) c;
            }
            throw new InputMismatchException();
        }

        public String nextToken() {
            int c = next();
            while (isSpaceChar(c))
                c = next();
            StringBuilder res = new StringBuilder();
            do {
                res.append((char) c);
                c = next();
            } while (!isSpaceChar(c));
            return res.toString();
        }

        public int nextInt() {
            int c = next();
            while (isSpaceChar(c))
                c = next();
            int sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = next();
            }
            int res = 0;
            do {
                if (c < '0' || c > '9')
                    throw new InputMismatchException();
                res *= 10;
                res += c-'0';
                c = next();
            } while (!isSpaceChar(c));
            return res*sgn;
        }

        public long nextLong() {
            int c = next();
            while (isSpaceChar(c))
                c = next();
            long sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = next();
            }
            long res = 0;
            do {
                if (c < '0' || c > '9')
                    throw new InputMismatchException();
                res *= 10;
                res += c-'0';
                c = next();
            } while (!isSpaceChar(c));
            return res*sgn;
        }

        public boolean isSpaceChar(int c) {
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }

        public interface SpaceCharFilter {
            public boolean isSpaceChar(int ch);
        }
    }

    static void debug(Object... o) {
        System.err.println(Arrays.deepToString(o));
    }
}