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;

class Program
{
    static void Main()
    {
        new Program().Solve();
    }

    long MOD = 1000000007L;

    void Solve()
    {
        long[] nmk = ReadLongArray();
        var n = nmk[0];
        var m = nmk[1];
        var k = nmk[2];

        var xy = 0L;

        for (var d = 1; d < n; d++)
        {
            xy += (n - d) * m % MOD * m % MOD * d % MOD;
            xy %= MOD;
        }

        for (var d = 1; d < m; d++)
        {
            xy += (m - d) * n % MOD * n % MOD * d % MOD;
            xy %= MOD;
        }

        Console.WriteLine(xy * Combination(n * m - 2, k - 2) % MOD);
    }

    /// <summary>
    /// 組合せ nCr = n! / r! / (n-r)!
    /// [フェルマーの小定理を使用] n! * r!^(m-2) * (n-r)!^(m-2)
    /// </summary>
    long Combination(long n, long r)
    {
        var p = new long[n + 1];
        p[0] = 1;
        p[1] = 1;
        for (var i = 2; i <= n; i++)
        {
            p[i] = i * p[i - 1] % MOD;
        }
        return p[n] * Power(p[r], MOD - 2) % MOD * Power(p[n - r], MOD - 2) % MOD;
    }

    /// <summary>
    /// べき乗 [バイナリ法/繰返し二乗法を使用]
    /// </summary>
    long Power(long a, long n)
    {
        var x = 1L;
        while (n > 0)
        {
            if ((n & 1) != 0) x = x * a % MOD;
            n >>= 1;
            a = a * a % MOD;
        }
        return x;
    }

    #region Console.Read 関係
    long[] ReadLongArray() => Array.ConvertAll(Console.ReadLine().Split(), long.Parse);
    long ReadLong() => long.Parse(Console.ReadLine());
    #endregion
}