Class Solution
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- All Implemented Interfaces:
public final class Solution3426 - Manhattan Distances of All Arrangements of Pieces.
Hard
You are given three integers
m,n, andk.There is a rectangular grid of size
m × ncontainingkidentical 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
kpieces on the grid with at most one piece per cell.Since the answer may be very large, return it modulo <code>10<sup>9</sup> + 7</code>.
The Manhattan Distance between two cells <code>(x<sub>i</sub>, y<sub>i</sub>)</code> and <code>(x<sub>j</sub>, y<sub>j</sub>)</code> is <code>|x<sub>i</sub> - x<sub>j</sub>| + |y<sub>i</sub> - y<sub>j</sub>|</code>.
Example 1:
Input: m = 2, n = 2, k = 2
Output: 8
Explanation:
The valid arrangements of pieces on the board are:
In the first 4 arrangements, the Manhattan distance between the two pieces is 1.
In the last 2 arrangements, the Manhattan distance between the two pieces is 2.
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 first and last arrangements have a total Manhattan distance of
1 + 1 + 2 = 4.The middle two arrangements have a total Manhattan distance of
1 + 2 + 3 = 6.
The total Manhattan distance between all pairs of pieces across all arrangements is
4 + 6 + 6 + 4 = 20.Constraints:
<code>1 <= m, n <= 10<sup>5</sup></code>
<code>2 <= m * n <= 10<sup>5</sup></code>
2 <= k <= m * n