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.Linq;
using System.Web;

namespace Competitive
{
    internal class Solution
    {
        public int N, M, K;
        public long MOD = 1000000007;
        public FermatCombination com;
        public void Run()
        {
            com = new FermatCombination(210000);
            {
                var line = Input.ReadIntArray();
                N = line[0];
                M = line[1];
                K = line[2];
            }

            long num = com.Combination(N * M, K);
            num %= MOD;
            long combpiece = com.Combination(K, 2);

            long x = avgsum(N) * M * M + avgsum(M) * N * N;
            x %= MOD;

            long divisor1 = (N * M) - 1;
            long divisor2 = (N * M);

            // 解答の計算
            long ret = 1;
            ret *= num;
            ret %= MOD;
            ret *= combpiece;
            ret %= MOD;

            ret *= x;
            ret %= MOD;
            ret *= div(divisor1);
            ret %= MOD;
            ret *= div(divisor2);
            ret %= MOD;
            Console.WriteLine(ret);
        }

        long avgsum(int length)
        {
            long ret = 0;
            for (int i = 0; i < length; i++)
            {
                long left = i;
                long right = length - 1 - i;
                ret += left * (left + 1) / 2L;
                ret += right * (right + 1) / 2L;
            }
            return ret;
        }

        long div(long divisor)
        {
            long ret = 1;
            ret *= com.Inverse[divisor];
            ret %= MOD;
            ret *= com.Factrial[divisor - 1];
            ret %= MOD;
            return ret;
        }
    }

    // libs ----------
    // フェルマーの定理、逆元を使う場合
    class FermatCombination
    {
        public long[] Factrial; // 階乗
        public long[] Inverse; // 逆元
        public long MOD = 1000000007;

        public FermatCombination(int n)
        {

            Factrial = new long[n + 1];
            Inverse = new long[n + 1];
            Factrial[0] = 1;
            Inverse[0] = 1;

            for (int i = 1; i <= n; i++)
            {
                Factrial[i] = (Factrial[i - 1] * i) % MOD;
            }

            for (int i = 1; i <= n; i++)
            {
                // フェルマーの小定理で逆元を求める
                Inverse[i] = Power(Factrial[i], (int)MOD - 2, MOD) % MOD;
            }
        }

        public long Permutation(int n, int k)
        {
            return Factrial[n] * Inverse[n - k] % MOD;
        }

        public long Combination(int n, int k)
        {
            return Factrial[n] * Inverse[k] % MOD * Inverse[n - k] % MOD;
        }

        public static long Power(long x, long n, long M)
        {
            long tmp = 1;

            if (n > 0)
            {
                tmp = Power(x, n / 2, M);
                if (n % 2 == 0) tmp = (tmp * tmp) % M;
                else tmp = (((tmp * tmp) % M) * x) % M;
            }

            return tmp;
        }
    }

    // common ----------

    internal static class Input
    {
        public static string ReadString()
        {
            string line = Console.ReadLine();
            return line;
        }

        public static int ReadInt()
        {
            string line = Console.ReadLine();
            return int.Parse(line);
        }

        public static int[] ReadIntArray()
        {
            string line = Console.ReadLine();
            return line.Split(' ').Select(int.Parse).ToArray();            
        }

        public static long ReadLong()
        {
            string line = Console.ReadLine();
            return long.Parse(line);
        }

        public static long[] ReadLongArray()
        {
            string line = Console.ReadLine();
            return line.Split(' ').Select(long.Parse).ToArray();
        }

        public static double[] ReadDoubleArray()
        {
            string line = Console.ReadLine();
            return line.Split(' ').Select(double.Parse).ToArray();
        }
    }
    
    internal class Program
    {
        public static void Main(string[] args)
        {
            var s = new Solution();
            s.Run();
        }
    }
}