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.Collections;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace AtCorder
{
	public class Program
	{
		const long MOD = 1000000007;
		static void Main(string[] args)
		{
			var cin = new Scanner2();
			long n = cin.Long();
			long m = cin.Long();
			long k = cin.Long();

			ModCounting.InitializeFactorial(n * m);

			long ans = 0;
			for (long i = 1; i <= m + n - 2; i++) {
				for (long j = 0; j <= i; j++) {
					long a = j;
					long b = i - j;
					long temp = (n - a) * (m - b);
					if (temp <= 0) {
						continue;
					}

					if (a != 0 && b != 0) {
						temp = (temp * 2) % MOD;
					}

					temp = (temp * i) % MOD;
					ans = (ans + temp) % MOD;
				}
			}

			long mul = ModCounting.Combination(m * n - 2, k - 2);
			ans = (ans * mul) % MOD;

			Console.WriteLine(ans);
		}
	}

	public class ModCounting
	{
		private static long p_;

		private static long[] factorial_;
		private static long[] inverseFactorial_;
		private static long[] inverse_;

		public static void InitializeFactorial(long max, long p = 1000000007)
		{
			p_ = p;

			factorial_ = new long[max + 1];
			inverseFactorial_ = new long[max + 1];
			inverse_ = new long[max + 1];

			factorial_[0] = factorial_[1] = 1;
			inverseFactorial_[0] = inverseFactorial_[1] = 1;
			inverse_[1] = 1;
			for (int i = 2; i <= max; i++) {
				factorial_[i] = factorial_[i - 1] * i % p;
				inverse_[i] = p - inverse_[p % i] * (p / i) % p;
				inverseFactorial_[i] = inverseFactorial_[i - 1] * inverse_[i] % p;
			}
		}

		public static long Factorial(long n)
		{
			if (n < 0) {
				return 0;
			}

			return factorial_[n];
		}

		public static long InverseFactorial(long n)
		{
			if (n < 0) {
				return 0;
			}

			return inverseFactorial_[n];
		}

		public static long Inverse(long n)
		{
			if (n < 0) {
				return 0;
			}

			return inverse_[n];
		}

		public static long Permutation(long n, long k)
		{
			if (n < k || (n < 0 || k < 0)) {
				return 0;
			}

			return factorial_[n] * inverseFactorial_[n - k] % p_;
		}

		public static long Combination(long n, long k)
		{
			if (n < k || (n < 0 || k < 0)) {
				return 0;
			}

			return factorial_[n] * (inverseFactorial_[k] * inverseFactorial_[n - k] % p_) % p_;
		}
	}

	public class Scanner2
	{
		private readonly char[] delimiter_ = new char[] { ' ' };
		private readonly string filePath_;
		private string[] buf_;
		private int index_;
		Func<string> reader_;

		public Scanner2(string file = "")
		{
			if (string.IsNullOrWhiteSpace(file)) {
				reader_ = Console.ReadLine;
			} else {
				filePath_ = file;
				var fs = new StreamReader(file);
				reader_ = fs.ReadLine;
			}
			buf_ = new string[0];
			index_ = 0;
		}

		public string Next()
		{
			if (index_ < buf_.Length) {
				return buf_[index_++];
			}

			string st = reader_();
			while (st == "") {
				st = reader_();
			}

			buf_ = st.Split(delimiter_, StringSplitOptions.RemoveEmptyEntries);
			if (buf_.Length == 0) {
				return Next();
			}

			index_ = 0;
			return buf_[index_++];
		}

		public int Int() => int.Parse(Next());
		public long Long() => long.Parse(Next());
		public double Double() => double.Parse(Next());

		public int[] ArrayInt(int N, int add = 0)
		{
			int[] Array = new int[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Int() + add;
			}
			return Array;
		}

		public long[] ArrayLong(int N, long add = 0)
		{
			long[] Array = new long[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Long() + add;
			}
			return Array;
		}

		public double[] ArrayDouble(int N, double add = 0)
		{
			double[] Array = new double[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Double() + add;
			}
			return Array;
		}

		public void Save(string text)
		{
			if (string.IsNullOrWhiteSpace(filePath_)) {
				return;
			}

			File.WriteAllText(filePath_ + "_output.txt", text);
		}
	}
}