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 static Input;
public class Hello
{
    private const long INF = 1000000007;
    private struct Pair { public long x, y; public Pair(long xx, long yy) { x = xx; y = yy; } }
    public static void Main()
    {
        int N = NextInt();
        int M = NextInt();
        int K = NextInt();
        //2点間さえわかればおｋ
        long ans = 0;
        long C = PCH.nCr(N * M - 2, K - 2);
        long[] X = new long[M + 1];
        for (int i = 0; i <= M; i++) { X[i] = i; X[i] %= INF; }
        for (int i = 1; i <= M; i++) { X[i] += X[i - 1]; X[i] %= INF; }

        long[] Y = new long[N + 1];
        for (int i = 0; i <= N; i++) { Y[i] = i * M; Y[i] %= INF; }
        for (int i = 1; i <= N; i++) { Y[i] += Y[i - 1]; Y[i] %= INF; }

        for (int x = 1; x <= M; x++)
        {
            for (int y = 1; y <= N; y++)
            {
                //Y座標は上にしか存在しない(同じ列のY座標はなので)
                ans += (((Y[N - y] + INF) % INF) * C) % INF;

                //X座標は絶えず変動するので自身より右、左側(領域高さ-1)を足す
                ans += (((X[M - x] + INF) % INF) * (N - y + 1) + ((X[x - 1] + INF) % INF) * (N - y)) * C % INF;
            }
        }
        Console.WriteLine(ans % INF);
    }
}
public static class PCH
{
    public static long nPr(long n, long r, long m = 1000000007)
    {
        for (long i = 0; i < r; r++) { n *= (n - i); n %= m; }
        return n;
    }
    public static long nCr(long n, long r, long m = 1000000007)
    {
        long a = 1;
        for (long i = 0; i < r; i++) { a *= (n - i); a %= m; }
        long b = 1;
        for (long x = 2; x <= r; x++) { b *= x; b %= m; }
        b = RepPow(b, m - 2) % m;
        return (a * b) % m;
    }
    public static long nHr(long n, long r, long m = 1000000007)
    {
        return nCr(n + r - 1, r, m);
    }
    public static long RepPow(long x, long p, long m = 1000000007)
    {
        if (p == 1) return x % m;
        long b = RepPow(x, p / 2, m) % m;
        if (p % 2 == 0) { return (b * b) % m; }
        else { return (x * ((b * b) % m)) % m; }
    }
}
public class Input
{
    private static Queue<string> queue = new Queue<string>();
    private static void Confirm() { if (queue.Count == 0) foreach (var s in Console.ReadLine().Split(' ')) queue.Enqueue(s); }
    public static int NextInt() { Confirm(); return int.Parse(queue.Dequeue()); }
    public static long NextLong() { Confirm(); return long.Parse(queue.Dequeue()); }
    public static string NextString() { Confirm(); return queue.Dequeue(); }
    public static double NextDouble() { Confirm(); return double.Parse(queue.Dequeue()); }
    public static int[] LineInt() { return Console.ReadLine().Split(' ').Select(int.Parse).ToArray(); }
    public static long[] LineLong() { return Console.ReadLine().Split(' ').Select(long.Parse).ToArray(); }
    public static string[] LineString() { return Console.ReadLine().Split(' ').ToArray(); }
    public static double[] LineDouble() { return Console.ReadLine().Split(' ').Select(double.Parse).ToArray(); }
}