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 PyPy2 (5.6.0) (judge verdict: Time Limit Exceeded)

MAX = 10**9 + 7

def mod_inv(i, MOD):
    return pow(i, MOD-2, MOD)

def combination(_N, _K):
    if _K > (_N/2):
        _K = _N - _K
    up = 1
    low = 1
    for i in range (_K):
        up *= (_N - i) % MAX
        low *= (i + 1) % MAX
    ret = up * mod_inv(low, MAX)
    if ret >= MAX:
        ret %= MAX
    return ret

def solve():
    N, M, K = [int(_) for _ in raw_input().split()]
    disSum = M*N* (M*(N+1)*(N-1) + N*(M+1)*(M-1)) / 6
    if disSum >= MAX:
        disSum %= MAX
    ret = disSum * combination(M*N-2, K-2)
    if ret >= MAX:
        ret %= MAX
    print ret

if __name__ == '__main__':
    solve()