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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

import math
N,M,K=map(int,input().split())
xs=0
ys=0
inf=10**9+7
xs=(M*(M+1)*(2*M+1) - 3*M*(M+1))//12
ys=(N*(N+1)*(2*N+1) - 3*N*(N+1))//12
#for i in range(1,M):
	#xs+=i*(M-i)
	#xs = xs % inf
#for i in range(1,N):
	#ys+=i*(N-i)
	#ys = ys % inf
xs*=(N**2)
xs = xs % inf
ys*=(M**2)
ys = ys % inf

def comb(n,r):
	return math.factorial(n) // (math.factorial(n-r)*math.factorial(r))

def comb_new(n,r):
	y = 1
	for i in range(r):
		y *= (n-i)
	return y//math.factorial(r)

#c = comb(N*M-2,K-2) % inf
c = comb_new(N*M-2,K-2) % inf
print(((xs+ys)*c)%inf)