Manhattan Distances of All Arrangements of Pieces

Description

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

There is a rectangular grid of size m × n containing k identical pieces. Return the sum of Manhattan distances between every pair of pieces over all valid arrangements of pieces.

A valid arrangement is a placement of all k pieces on the grid with at most one piece per cell.

Since the answer may be very large, return it modulo 109 + 7.

The Manhattan Distance between two cells (xi, yi) and (xj, yj) is |xi - xj| + |yi - yj|.