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)

mdl = 10**9 + 7

N, M, K = map(int,input().split())

nm = N*M
mm = M*M % mdl
nn = N*N % mdl

U = 2*10**5
fact = [1] * (U+1)

for i in range(1,U+1):
    fact[i] = (fact[i-1]*i) % mdl

def comb(n,k):
    x = fact[n]
    x *= pow(fact[n-k], mdl-2, mdl)
    x %= mdl
    x *= pow(fact[k], mdl-2, mdl)
    x %= mdl
    return x

def F(N):
    x = N*(N+1) // 2 * N
    y = N*(N+1)*(2*N+1) // 6
    return (x-y) % mdl

ans = mm*F(N) % mdl + nn*F(M) % mdl
ans %= mdl
ans *= comb(nm-2,K-2)
ans %= mdl

print(ans)