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: Runtime Error)

using System;
using System.Collections.Generic;
using System.Collections;
using System.Collections.Specialized;
using System.Linq;
using System.Text;
using System.IO;
static class Program{
	const int mod=(int)1e9+7;
	static void Main(){
		Sc sc=new Sc();
		int n=sc.I;
		List<int> li=new List<int>();
		li.Add((int)-1);
		li.Add((int)2e9+2);
		long cs=0,c1=0,t2=0,cc=0;
		StringBuilder sb=new StringBuilder();
		for(int i = 0,p=0;i<n;i++) {
			int[] e=sc.Ia;
			if(e[0]==1){
				p++;
				t2+=e[2];
				int lb=0,ub=li.Count-1,mid=0;
				while(ub-lb>1){
					mid=(ub+lb)/2;
					if(li[mid]>=e[1]){ub=mid;}
					else{lb=mid;}
				}
				//li.Insert(ub,e[1]+(int)1e9);
				e[1]+=+(int)1e9;
				li.Insert(ub,e[1]);
				cs+=e[1];
				if(i==0){ }
				else if(cc<li[(li.Count-1)>>1]){c1+=cc;}
				if(e[1]<li[(li.Count-1)>>1]){c1+=cc;}
				cc=li[(li.Count-1)>>1];
			}
			else{
				long ans=cs-((long)li[(li.Count-1)>>1])*p+t2;
				sb.Append((li[(li.Count-1)>>1]-(int)1e9)+" "+ans+"\n");
			}
		}
		Console.Write(sb);
	}
}

public class Sc{
	private string[] a;
	private int n=-1,m=0;
	public int I{get{if(n==-1||n==a.Length-1){n=-1;a=Console.ReadLine().Split();}n++;return int.Parse(a[n]);}}
	public long L{get{if(n==-1||n==a.Length-1){n=-1;a=Console.ReadLine().Split();}n++;return long.Parse(a[n]);}}
	public double D{get{if(n==-1||n==a.Length-1){n=-1;a=Console.ReadLine().Split();}n++;return double.Parse(a[n]);}}
	public string S{get{if(n==-1||n==a.Length-1){n=-1;a=Console.ReadLine().Split();}n++;return a[n];}}
	public string Sz{get{return Console.ReadLine();}}
	public int[] Ia{get{if(n!=-1&&n!=a.Length-1){m=n;n=-1;return Array.ConvertAll(a.Skip(m+1).ToArray(),int.Parse);}return Array.ConvertAll(Console.ReadLine().Split(),int.Parse);}}
	public int[] Ia2(int b){return Array.ConvertAll((((b&1)!=0?"0 ":"")+Console.ReadLine()+((b&2)!=0?" 0":"")).Split(),int.Parse);}
	public long[] La{get{return Array.ConvertAll(Console.ReadLine().Split(),long.Parse);}}
	public long[] La2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),long.Parse);}}
	public double[] Da{get{return Array.ConvertAll(Console.ReadLine().Split(),double.Parse);}}
	public string[] Sa{get{return Console.ReadLine().Split();}}
	public string[] Sa2{get{return ("# "+Console.ReadLine()).Split();}}
}