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.Linq;
using System.Collections.Generic;
using static System.Console;
using System.Text;
namespace AtCoder_Beginner_Contest_127
{
    class Program
    {
        public class Combination
        {
            readonly long mod;
            readonly long[] fac, finv, inv;
            public Combination(long n, long mod = 1000000007)
            {
                long max = n + 1;
                this.mod = mod;
                fac = new long[max];
                finv = new long[max];
                inv = new long[max];
                fac[0] = fac[1] = 1;
                finv[0] = finv[1] = 1;
                inv[1] = 1;
                for (int i = 2; i < max; ++i)
                {
                    fac[i] = fac[i - 1] * i % mod;
                    inv[i] = mod - inv[mod % i] * (mod / i) % mod;
                    finv[i] = finv[i - 1] * inv[i] % mod;
                }
            }
            public long com(int n, int k)
            {
                if (n < k) return 0;
                if (n < 0 | k < 0) return 0;
                return fac[n] * (finv[k] * finv[n - k] % mod) % mod;
            }
        }
        static public List<int> SIn() { return ReadLine().Split().Select(int.Parse).ToList(); }
        static void Main(string[] args)
        {
            var NMK = SIn();
            var N = NMK[0];//N行
            var M = NMK[1];//M列
            var K = NMK[2];//K個

            long mod = 1000000000 + 7;
            long nsum = 0;
            for (int i = 1; i <= N - 1; ++i)
            {
                nsum = nsum + (long) i * (N - i) % mod;
                nsum %= mod;
            }
            long msum = 0;
            for (int i = 1; i <= M - 1; ++i)
            {
                msum = msum + (long)i * (M - i) % mod;
                msum %= mod;
            }
            long sum = (nsum * (M * M % mod)) % mod + (msum * (N * N % mod)) % mod;
            sum %= mod;
            var cmb = new Combination(N * M - 2, mod);
            sum = sum * cmb.com(N * M - 2, K - 2) % mod;
            WriteLine(sum);
        }
    }
}