How to Solve Manhattan Distances of All Arrangements of Pieces Problem
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Manhattan Distances of All Arrangements of Pieces
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 arr...
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Understanding the Manhattan Distances of All Arrangements of Pieces Problem
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Problem Statement
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|.
Manhattan Distances of All Arrangements of Pieces
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Examples
m = 2, n = 2, k = 28The 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.
m = 1, n = 4, k = 320The 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
1 <= m, n <= 1052 <= m * n <= 1052 <= k <= m * nHow Interview Coder Helps with Leetcode Problems
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