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.

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.

Constraints

• $2 \leq N \times M \leq 2 \times 10^5$
• $2 \leq K \leq N \times M$
• All values in input are integers.

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.

见题面 Input 部分

见题面 Output 部分