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.util.Scanner;
public class Main{
	public static void main(String[] args){
		Scanner sc = new Scanner(System.in);
		int n=sc.nextInt();
		int m=sc.nextInt();
		int k=sc.nextInt();
		long mod=1000000007;
		long count=0;
		for(int i=1; i<=m; i++){
			count+=((n*n)%mod)*((m-i+1)*(m-i)/2)%mod;
			count%=mod;
		}
		for(int i=1; i<=n; i++){
			count+=((m*m)%mod)*((n-i+1)*(n-i)/2)%mod;
			count%=mod;
		}
		System.out.println((count*Combination(n*m-2,k-2,mod))%mod);
	}
	static long Combination(int left,int right,long mod){
		long ans=1;
		right=Math.min(right,left-right);
		for(int i=0; i<right; i++){
			ans*=(left-i);
			ans%=mod;
		}
		for(int i=2; i<=right; i++){
			long tmp=mod-2;
			long mul=i;
			while(tmp>0){
				if((tmp&1)==1){
					ans*=mul;
				}
				tmp>>=1;
				mul*=mul;
				ans%=mod;
				mul%=mod;
			}
		}
		return ans;
	}
}