Problem:
Score : 500 points

Problem Statement

We have a grid of squares with N rows and M columns. Let (i, j) denote the square at the i-th row from the top and j-th column from the left. We will choose K of the squares and put a piece on each of them.

If we place the K pieces on squares (x_1, y_1), (x_2, y_2), ..., and (x_K, y_K), the cost of this arrangement is computed as:

\sum_{i=1}^{K-1} \sum_{j=i+1}^K (|x_i - x_j| + |y_i - y_j|)

Find the sum of the costs of all possible arrangements of the pieces. Since this value can be tremendous, print it modulo 10^9+7.

We consider two arrangements of the pieces different if and only if there is a square that contains a piece in one of the arrangements but not in the other.

Constraints

2 \leq N \times M \leq 2 \times 10^5

2 \leq K \leq N \times M

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M K

Output

Print the sum of the costs of all possible arrangements of the pieces, modulo 10^9+7.

Sample Input 1

2 2 2

Sample Output 1

8

There are six possible arrangements of the pieces, as follows:

((1,1),(1,2)), with the cost |1-1|+|1-2| = 1

((1,1),(2,1)), with the cost |1-2|+|1-1| = 1

((1,1),(2,2)), with the cost |1-2|+|1-2| = 2

((1,2),(2,1)), with the cost |1-2|+|2-1| = 2

((1,2),(2,2)), with the cost |1-2|+|2-2| = 1

((2,1),(2,2)), with the cost |2-2|+|1-2| = 1

The sum of these costs is 8.

Sample Input 2

4 5 4

Sample Output 2

87210

Sample Input 3

100 100 5000

Sample Output 3

817260251

Be sure to print the sum modulo 10^9+7.

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

using System;
using System.Collections.Generic;
using System.IO;
using System.Runtime.Remoting.Contexts;
using Microsoft.Win32;

namespace Contest
{
    class Program
    {
        private Scanner sc;
        private const int Mod = (int)1e9 + 7;
        private long N, M, K;

        public void Solve()
        {
            sc = new Scanner();
            N = sc.NextInt();
            M = sc.NextInt();
            K = sc.NextLong();

            ModInt ans = 0;

            ModInt n = 0;
            for (int i = 0; i < N; i++)
            {
                n += i;
                ans += n * M * M;
            }

            ModInt m = 0;
            for (int i = 0; i < M; i++)
            {
                m += i;
                ans += m * N * N;
            }

            ModInt a = 1;
            ModInt b = 1;
            for (int i = 1; i <= K - 2; i++)
            {
                a *= i;
                b *= N * M - 2 - i + 1;
            }


            // ans * (N*M-2) C K-2
            Console.WriteLine(ans.Num * b / a);


        }

        static void Main() => new Program().Solve();
    }
}

struct ModInt
{
    public long Num;
    public const long Mod = (long)1e9 + 7;
    public ModInt(long n) { Num = n % Mod; }

    public override string ToString() { return Num.ToString(); }
    public override bool Equals(object b) { return Num.Equals(b); }

    public static ModInt operator +(ModInt a, ModInt b)
    {
        return (a.Num + b.Num) % Mod;
    }
    public static ModInt operator -(ModInt a, ModInt b)
    {
        return (a.Num - b.Num) % Mod;
    }
    public static ModInt operator *(ModInt a, ModInt b)
    {
        return (a.Num * b.Num) % Mod;
    }
    public static ModInt operator %(ModInt a, long b)
    {
        return ((a.Num % b) + Mod) % Mod;
    }
    public static ModInt operator /(ModInt a, ModInt b)
    {
        return a * ModInt.Pow(b, Mod - 2);
    }

    public static ModInt Pow(ModInt a, long b)
    {
        if (b == 0) return 1;
        return (b % 2 == 1 ? a : 1) * Pow(a * a, b / 2);
    }

    public static implicit operator ModInt(long n) { return new ModInt(n); }


}

class Scanner
{
    public Scanner()
    {
        _stream = new StreamReader(Console.OpenStandardInput());
        _pos = 0;
        _line = new string[0];
        _separator = ' ';
    }
    private readonly char _separator;
    private readonly StreamReader _stream;
    private int _pos;
    private string[] _line;
    #region get a element
    public string Next()
    {
        if (_pos >= _line.Length)
        {
            _line = _stream.ReadLine().Split(_separator);
            _pos = 0;
        }
        return _line[_pos++];
    }
    public int NextInt()
    {
        return int.Parse(Next());
    }
    public long NextLong()
    {
        return long.Parse(Next());
    }
    public double NextDouble()
    {
        return double.Parse(Next());
    }
    #endregion
    #region convert array
    private int[] ToIntArray(string[] array)
    {
        var result = new int[array.Length];
        for (int i = 0; i < array.Length; i++)
        {
            result[i] = int.Parse(array[i]);
        }

        return result;
    }
    private long[] ToLongArray(string[] array)
    {
        var result = new long[array.Length];
        for (int i = 0; i < array.Length; i++)
        {
            result[i] = long.Parse(array[i]);
        }

        return result;
    }
    private double[] ToDoubleArray(string[] array)
    {
        var result = new double[array.Length];
        for (int i = 0; i < array.Length; i++)
        {
            result[i] = double.Parse(array[i]);
        }

        return result;
    }
    #endregion
    #region get array
    public string[] Array()
    {
        if (_pos >= _line.Length)
            _line = _stream.ReadLine().Split(_separator);
        _pos = _line.Length;
        return _line;
    }
    public int[] IntArray()
    {
        return ToIntArray(Array());
    }
    public long[] LongArray()
    {
        return ToLongArray(Array());
    }
    public double[] DoubleArray()
    {
        return ToDoubleArray(Array());
    }
    #endregion
}