Cell Distance

AtCoder

IDabc127_e

Time2000ms

Memory256MB

Difficulty—

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. If 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: $\sum_{i=1}^{K-1} \sum_{j=i+1}^K (|x_i - x_j| + |y_i - y_j|)$ Find the sum of the costs of all possible arrangements of the pieces. Since this value can be tremendous, print it modulo $10^9+7$. We 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. ## Constraints * $2 \leq N \times M \leq 2 \times 10^5$ * $2 \leq K \leq N \times M$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $K$ [samples]

Samples

Input #1

2 2 2

Output #1

8

There are six possible arrangements of the pieces, as follows:

*   $((1,1),(1,2))$, with the cost $|1-1|+|1-2| = 1$
*   $((1,1),(2,1))$, with the cost $|1-2|+|1-1| = 1$
*   $((1,1),(2,2))$, with the cost $|1-2|+|1-2| = 2$
*   $((1,2),(2,1))$, with the cost $|1-2|+|2-1| = 2$
*   $((1,2),(2,2))$, with the cost $|1-2|+|2-2| = 1$
*   $((2,1),(2,2))$, with the cost $|2-2|+|1-2| = 1$

The sum of these costs is $8$.

Input #2

4 5 4

Output #2

Input #3

100 100 5000

Output #3

817260251

Be sure to print the sum modulo $10^9+7$.

API Response (JSON)

{
  "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 pi...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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