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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Linq.Expressions;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Number = System.Int64;
namespace Program
{
    public class Solver
    {
        public void Solve()
        {
            var n = sc.Integer();
            var A = new int[n];
            var B = new int[n];
            const int ADD = 2100;
            var dp = Enumerate(4200, x => new ModInteger[4200]);
            for (int i = 0; i < n; i++)
            {
                A[i] = sc.Integer();
                B[i] = sc.Integer();
                dp[ADD - A[i]][ADD - B[i]] += 1;
            }
            for (int i = 1; i < 4200; i++)
                for (int j = 1; j < 4200; j++)
                    dp[i][j] += dp[i - 1][j] + dp[i][j - 1];
            ModInteger sum = 0;
            for (int i = 0; i < n; i++)
                sum += dp[ADD + A[i]][ADD + B[i]];


            var fact = new ModInteger[8200];
            var inv = new ModInteger[8200];
            fact[0] = inv[0] = 1;
            for (int i = 1; i < 8200; i++)
            {
                fact[i] = fact[i - 1] * i;
                inv[i] = ModInteger.Inverse(fact[i]);
            }
            for (int i = 0; i < n; i++)
                sum -= fact[2 * A[i] + 2 * B[i]] * inv[2 * A[i]] * inv[2 * B[i]];


            sum *= ModInteger.Inverse(2);
            IO.Printer.Out.WriteLine(sum);


        }

        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());
        static T[] Enumerate<T>(int n, Func<int, T> f) { var a = new T[n]; for (int i = 0; i < n; ++i) a[i] = f(i); return a; }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }


    }
}
#region main
static class Ex
{
    static public string AsString(this IEnumerable<char> ie) { return new string(System.Linq.Enumerable.ToArray(ie)); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") { return string.Join(st, ie); }
    static public void Main()
    {
        var solver = new Program.Solver();
        solver.Solve();
        Program.IO.Printer.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO
{
    using System.IO;
    using System.Text;
    using System.Globalization;
    public class Printer: StreamWriter
    {
        static Printer() { Out = new Printer(Console.OpenStandardOutput()) { AutoFlush = false }; }
        public static Printer Out { get; set; }
        public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
        public Printer(System.IO.Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
        public Printer(System.IO.Stream stream, Encoding encoding) : base(stream, encoding) { }
        public void Write<T>(string format, T[] source) { base.Write(format, source.OfType<object>().ToArray()); }
        public void WriteLine<T>(string format, T[] source) { base.WriteLine(format, source.OfType<object>().ToArray()); }
    }
    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }
        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) return 0;
            if (ptr >= len) { ptr = 0; if ((len = str.Read(buf, 0, 1024)) <= 0) { isEof = true; return 0; } }
            return buf[ptr++];
        }
        public char Char() { byte b = 0; do b = read(); while ((b < 33 || 126 < b) && !isEof); return (char)b; }

        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n'; b = (char)read())
                if (b == 0) break;
                else if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long()
        {
            if (isEof) return long.MinValue;
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != 0 && b != '-' && (b < '0' || '9' < b));
            if (b == 0) return long.MinValue;
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        public int Integer() { return (isEof) ? int.MinValue : (int)Long(); }
        public double Double() { var s = Scan(); return s != "" ? double.Parse(s, CultureInfo.InvariantCulture) : double.NaN; }
        private T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public char[] Char(int n) { return enumerate(n, Char); }
        public string[] Scan(int n) { return enumerate(n, Scan); }
        public double[] Double(int n) { return enumerate(n, Double); }
        public int[] Integer(int n) { return enumerate(n, Integer); }
        public long[] Long(int n) { return enumerate(n, Long); }
    }
}
#endregion
#region ModNumber
public partial struct ModInteger
{
    public const long Mod = (long)1e9 + 7;
    public long num;
    public ModInteger(long n) : this() { num = n % Mod; if (num < 0) num += Mod; }
    public override string ToString() { return num.ToString(); }
    public static ModInteger operator +(ModInteger l, ModInteger r) { var n = l.num + r.num; if (n >= Mod) n -= Mod; return new ModInteger() { num = n }; }
    public static ModInteger operator -(ModInteger l, ModInteger r) { var n = l.num + Mod - r.num; if (n >= Mod) n -= Mod; return new ModInteger() { num = n }; }
    public static ModInteger operator *(ModInteger l, ModInteger r) { return new ModInteger(l.num * r.num); }
    public static ModInteger operator ^(ModInteger l, long r) { return ModInteger.Pow(l, r); }
    public static implicit operator ModInteger(long n) { return new ModInteger(n); }
    public static ModInteger Pow(ModInteger v, long k)
    {
        ModInteger ret = 1;
        var n = k;
        for (; n > 0; n >>= 1, v *= v)
        {
            if ((n & 1) == 1)
                ret = ret * v;
        }
        return ret;
    }
}
#endregion
#region Inverse
public partial struct ModInteger
{
    static public ModInteger Inverse(ModInteger v)
    {
        long p, q;
        ExGCD(v.num, Mod, out p, out q);
        return new ModInteger(p % Mod + Mod);
    }
    static public long ExGCD(long a, long b, out long x, out long y)
    {
        var u = new long[] { a, 1, 0 };
        var v = new long[] { b, 0, 1 };
        while (v[0] != 0)
        {
            var t = u[0] / v[0];
            for (int i = 0; i < 3; i++)
            {
                var tmp = u[i] - t * v[i];
                u[i] = v[i];
                v[i] = tmp;
            }
        }
        x = u[1];
        y = u[2];
        if (u[0] > 0)
            return u[0];
        for (int i = 0; i < 3; i++)
            u[i] = -u[i];
        return u[0];

    }
}
#endregion
#region ModTable
public class ModTable
{
    public ModInteger[] perm, inv;
    public ModTable(int n)
    {
        perm = new ModInteger[n + 1];
        inv = new ModInteger[n + 1];
        perm[0] = 1;
        for (int i = 1; i <= n; i++)
            perm[i] = perm[i - 1] * i;
        inv[n] = ModInteger.Inverse(perm[n]);
        for (int i = n - 1; i >= 0; i--)
            inv[i] = inv[i + 1] * (i + 1);
        inv[0] = inv[1];
    }
    public ModInteger Inverse(int k) { return inv[k]; }
    public ModInteger Permutation(int n, int k)
    {
        if (n < 0 || n >= perm.Length)
            return 0;
        if (k < 0 || k >= n)
            return 0;
        return perm[n] * inv[n - k];
    }
    public ModInteger Combination(int n, int r)
    {
        if (n < 0 || n >= perm.Length || r < 0 || r > n) return 0;
        return perm[n] * inv[n - r] * inv[r];
    }
    public ModInteger RepeatedCombination(int n, int k)
    {
        if (k == 0) return 1;
        return Combination(n + k - 1, k);
    }
}
#endregion