{"problem":{"name":"Cell Distance","description":{"content":"We have a grid of squares with $N$ rows and $M$ columns. Let $(i, j)$ denote the square at the $i$\\-th row from the top and $j$\\-th column from the left. We will choose $K$ of the squares and put a pi","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc127_e"},"statements":[{"statement_type":"Markdown","content":"We have a grid of squares with $N$ rows and $M$ columns. Let $(i, j)$ denote the square at the $i$\\-th row from the top and $j$\\-th column from the left. We will choose $K$ of the squares and put a piece on each of them.\nIf we place the $K$ pieces on squares $(x_1, y_1)$, $(x_2, y_2)$, ..., and $(x_K, y_K)$, the _cost_ of this arrangement is computed as:\n$\\sum_{i=1}^{K-1} \\sum_{j=i+1}^K (|x_i - x_j| + |y_i - y_j|)$\nFind the sum of the costs of all possible arrangements of the pieces. Since this value can be tremendous, print it modulo $10^9+7$.\nWe consider two arrangements of the pieces different if and only if there is a square that contains a piece in one of the arrangements but not in the other.\n\n## Constraints\n\n*   $2 \\leq N \\times M \\leq 2 \\times 10^5$\n*   $2 \\leq K \\leq N \\times M$\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$ $M$ $K$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc127_e","tags":[],"sample_group":[["2 2 2","8\n\nThere are six possible arrangements of the pieces, as follows:\n\n*   $((1,1),(1,2))$, with the cost $|1-1|+|1-2| = 1$\n*   $((1,1),(2,1))$, with the cost $|1-2|+|1-1| = 1$\n*   $((1,1),(2,2))$, with the cost $|1-2|+|1-2| = 2$\n*   $((1,2),(2,1))$, with the cost $|1-2|+|2-1| = 2$\n*   $((1,2),(2,2))$, with the cost $|1-2|+|2-2| = 1$\n*   $((2,1),(2,2))$, with the cost $|2-2|+|1-2| = 1$\n\nThe sum of these costs is $8$."],["4 5 4","87210"],["100 100 5000","817260251\n\nBe sure to print the sum modulo $10^9+7$."]],"created_at":"2026-03-03 11:01:14"}}