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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.io.*;
import java.util.*;





public class Main
{

	static OutputStream outputStream = System.out;
	static PrintWriter out = new PrintWriter(outputStream);	

	static int n,m;
	static long mod=1000000007;

	static long[] fact;

	static long power(long x,long y)
	{
		if(y==0) return 1;
		long cp = power(x,y/2);

		if(y%2==0) return (cp*cp)%mod;
		return ((cp*cp)%mod * x)%mod;
	}

	public static void main(String[] args)throws IOException
	{
		BufferedReader br=new BufferedReader(new InputStreamReader(System.in));

		// long n=Long.parseLong(br.readLine());


		String[] s=br.readLine().split(" ");

		n=Integer.parseInt(s[0]);m=Integer.parseInt(s[1]);
		int k=Integer.parseInt(s[2]);

		long dis=0;

		for(int i=0;i<n;i++)
		{
			for(int j=0;j<m;j++)
			{
				long left = j;

				left*=(left+1);
				left/=2;
				left*=n;

				dis+=(left)%mod;

				left = i;

				left*=(left+1);
				left/=2;
				left*=m;

				dis+=(left)%mod;

			}
		}

		fact=new long[n*m+10];
		fact[0]=1;

		for(int i=1;i<n*m+10;i++)
		{
			fact[i]=(i*fact[i-1])%mod;
		}

		long ans=fact[n*m-2];

		ans*=power(fact[k-2],mod-2);
		ans%=mod;

		ans*=power(fact[n*m-k],mod-2);
		ans%=mod;

		ans*=dis;
		ans%=mod;

		out.println(ans);

		out.close();

	}
}