You are given three integers m, n, and k.

Consider an m by n grid where k identical pieces are placed such that each cell contains at most $1$ piece.

The Manhattan distance between two cells at coordinates $(r_1, c_1)$ and $(r_2, c_2)$ is defined as $|r_1 - r_2| + |c_1 - c_2|$.

Return the total sum of Manhattan distances between all pairs of pieces across all valid arrangements. Since the answer may be very large, return it modulo $10^9 + 7$.

Constraints:

• $1 \leq$ m, n $\leq 10^5$

• $2 \leq$ m $ imes$ n $\leq 10^5$

• $2 \leq$ k $\leq$ m $ imes$ n

Now, let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

We have a game for you to play. Rearrange the logical building blocks to develop a clearer understanding of how to solve this problem.

Implement your solution in the following coding playground.