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: Wrong Answer)

from math import factorial
N, M, K = map(int, input().split())

MOD = 10**9+7

# WARN: Require mod number m is prime number
class ModComb:
    # For nCr mod m
    def __init__(self, n_max, m):
        self._n_max = n_max
        self._m = m
        self._mod_facts = [1] * (n_max+1)       # list of n! mod m
        self._mod_inv_facts = [1] * (n_max+1)   # list of inverse r! mod m

        if not m >= 3:
            raise ValueError('Require mod number m >= 3')

        # Calc n! mod m
        for i in range(2, n_max+1):
            self._mod_facts[i] = self._mod_facts[i-1] * i % m

        # Calc inverse r! mod m
        self._mod_inv_facts[n_max] = pow(self._mod_facts[n_max], m-2, m)
        for i in range(n_max-1, 1, -1):
            self._mod_inv_facts[i] = self._mod_inv_facts[i+1] * (i+1) % m

    def calc(self, n, r):
        # Return nCr mod m
        if n > self._n_max:
            raise Exception('n is larger than n_max')

        if r == 0:
            return 1

        m = self._m
        return self._mod_facts[n] * self._mod_inv_facts[r] * self._mod_inv_facts[n-r] % m

modcomb = ModComb(N*M-2, MOD)
p = modcomb.calc(N*M-2, K-2)
print(N*M-2, K-2, p)

ans = 0
for i in range(N):
    for j in range(M):
        if i == 0 and j == 0:
            continue
        d = i+j
        cnt = (N-i) * (M-j)
        if i != 0 and j != 0:
            cnt *= 2
        ans += d * cnt % MOD
ans = ans * p % MOD

print(ans)