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: Time Limit Exceeded)

using System;
using System.Numerics;

class E
{
	static void Main()
	{
		var a = Array.ConvertAll(Console.ReadLine().Split(), long.Parse);
		BigInteger h = a[0], w = a[1], k = a[2];

		Func<BigInteger, BigInteger, BigInteger> c = (x, y) => x * (x * x - 1) / 6 * y * y;
		Console.WriteLine((c(h, w) + c(w, h)) * Ncr(h * w - 2, k - 2) % 1000000007);
	}

	static BigInteger Factorial(BigInteger n) { for (BigInteger x = 1, i = 1; ; x *= ++i) if (i >= n) return x; }
	static BigInteger Npr(BigInteger n, BigInteger r)
	{
		if (n < r) return 0;
		for (BigInteger x = 1, i = n - r; ; x *= ++i) if (i >= n) return x;
	}
	static BigInteger Ncr(BigInteger n, BigInteger r) => n < r ? 0 : n - r < r ? Ncr(n, n - r) : Npr(n, r) / Factorial(r);
}