Manhattan Distances of All Arrangements of Pieces
You are given three integers m, n, and k.
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 10^9 + 7.
The Manhattan Distance between two cells (xi, yi) and (xj, yj) is |xi - xj| + |yi - yj|.
Example 1
Input: m = 2, n = 2, k = 2
Output: 8
Explanation: The valid arrangements of pieces on the board are: Thus, the total Manhattan distance across all valid arrangements is 1 + 1 + 1 + 1 + 2 + 2 = 8 .
Example 2
Input: m = 1, n = 4, k = 3
Output: 20
Explanation: The valid arrangements of pieces on the board are: The total Manhattan distance between all pairs of pieces across all arrangements is 4 + 6 + 6 + 4 = 20 .
Constraints
- 1 <= m, n <= 10^5
- 2 <= m * n <= 10^5
- 2 <= k <= m * n
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