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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;

public class Hello
{
    private const int MOD_N = 1000000000 + 7;

    public static void Main()
    {
        var list = Console.ReadLine().Split(' ');
        var n = int.Parse(list[0]);
        var m = int.Parse(list[1]);
        var k = int.Parse(list[2]);

        long total = 0;
        for (int i = 1; i < n; ++i)
        {
            for (int j = 1; j < m; ++j)
            {
                for (int a = i + 1; a <= n; ++a)
                {
                    for (int b = j + 1; b <= m; ++b)
                    {
                        total += a + b - i - j;
                        total %= MOD_N;
                    }
                }
            }
        }

        Console.WriteLine((total * Comb(n * m, k) % MOD_N));
    }

    private static int Comb(int m, int n)
    {
        var t = new int[m + 1];
        t[0] = t[1] = 1;

        for (int i = 1; i < m; ++i)
        {
            for (int j = i + 1; j != 0; --j)
                t[j] += t[j - 1];
        }

        return t[n];
    }
}