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.Generic;
using System.Linq;
using BitArray = System.Collections.BitArray;
using BigInteger = System.Numerics.BigInteger;

namespace AtCoderProject
{
    public class Program
    {
        public object Calc()
        {
            var line = consoleReader.SplitInt();
            var N = line[0];
            var M = line[1];
            var K = line[2];

            long sum = 0;
            for (int i = 1; i < N; i++)
            {
                sum = (sum + (long)i * (N - i) * M * M) % mod;
            }
            for (int i = 1; i < M; i++)
            {
                sum = (sum + (long)i * (M - i) * N * N) % mod;
            }

            var combination = Combination(N * M - 2, K - 2);

            return sum * combination % mod;
        }
        BigInteger Combination(int n, int k)
        {
            if (n / 2 < k) return Combination(n, n - k);

            BigInteger ret = 1;
            for (var i = 0; i < k; i++)
            {
                ret *= n - i;
                ret /= i + 1;
            }
            return ret;
        }

        public const int mod = 1000000007;
        #region いつもの
        private readonly ConsoleReader consoleReader;
        public Program() : this(new ConsoleReader()) { }
        public Program(ConsoleReader consoleReader) { this.consoleReader = consoleReader; }
        static void Main() => Console.WriteLine(new Program().Calc()); static string MultiLine<T>(IEnumerable<T> source) => string.Join("\n", source);
        #endregion
    }

    #region Console
    public class ConsoleReader { public virtual string String() => Console.ReadLine(); public int Int() => int.Parse(String()); public long Long() => long.Parse(String()); public ulong ULong() => ulong.Parse(String()); public double Double() => double.Parse(String()); public string[] Split() => String().Split().ToArray(); public int[] SplitInt() => String().Split().Select(int.Parse).ToArray(); public long[] SplitLong() => String().Split().Select(long.Parse).ToArray(); public ulong[] SplitULong() => String().Split().Select(ulong.Parse).ToArray(); public double[] SplitDouble() => String().Split().Select(double.Parse).ToArray(); public IEnumerable<string> Repeat(int count) { for (int i = 0; i < count; i++) yield return String(); } public IEnumerable<int> RepeatInt(int count) { for (int i = 0; i < count; i++) yield return Int(); } public IEnumerable<long> RepeatLong(int count) { for (int i = 0; i < count; i++) yield return Long(); } public IEnumerable<string[]> RepeatSplit(int count) { for (int i = 0; i < count; i++) yield return Split(); } public IEnumerable<int[]> RepeatSplitInt(int count) { for (int i = 0; i < count; i++) yield return SplitInt(); } }
    #endregion
}