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: Accepted)

import sys, math, collections, heapq, itertools
F = sys.stdin
def single_input(): return F.readline().strip("\n")
def line_input(): return F.readline().strip("\n").split()
mod = 7 + 10 ** 9
factorial = [1] * (2 * 10 ** 5 + 1)
for i in range(1, 2 * 10 ** 5 + 1):
    factorial[i] = (factorial[i-1] * i) % mod
revfactorial = [1] * (2 * 10 ** 5 + 1)
revfactorial[-1] = pow(factorial[-1], mod - 2, mod)
for i in reversed(range(1, 2 * 10**5)):
    revfactorial[i] = (revfactorial[i+1] * (i+1)) % mod

def comb(n, r):
    return (factorial[n] * revfactorial[n-r] * revfactorial[r]) % mod

def solve():
    N, M, K = map(int, line_input())
    xsum, ysum = 0, 0
    for i in range(1, N): #xsum. (pairの取り方)  * d
        xsum += ((N - i) * M * M * i) % mod
        xsum %= mod
    for j in range(1, M):
        ysum += ((M - j) * N * N * j) % mod
        ysum %= mod
    print(((xsum + ysum) * comb(N*M-2, K-2)) % mod)
    return 0
  
if __name__ == "__main__":
    solve()