Paper Cutting

AtCoder

IDkeyence2019_f

Time2000ms

Memory256MB

Difficulty—

There is a rectangular sheet of paper with height $H+1$ and width $W+1$. We introduce an $xy$\-coordinate system so that the four corners of the sheet are $(0, 0)$, $(W + 1, 0)$, $(0, H + 1)$ and $(W + 1, H + 1)$. This sheet can be cut along the lines $x = 1,2,...,W$ and the lines $y = 1,2,...,H$. Consider a sequence of operations of length $K$ where we choose $K$ of these $H + W$ lines and cut the sheet along those lines one by one in some order. Let the score of a cut be the number of pieces of paper that exist just after the cut. The score of a sequence of operations is the sum of the scores of all of the $K$ cuts. Find the sum of the scores of all possible sequences of operations of length $K$. Since this value can be extremely large, print the number modulo $10^9 + 7$. ## Constraints * $1 \leq H,W \leq 10^7$ * $1 \leq K \leq H + W$ * $H, W$ and $K$ are integers. ## Input Input is given from Standard Input in the following format: $H$ $W$ $K$ [samples]

Samples

Input #1

2 1 2

Output #1

34

Let $x_1$, $y_1$ and $y_2$ denote the cuts along the lines $x = 1$, $y = 1$ and $y = 2$, respectively. The six possible sequences of operations and the score of each of them are as follows:

*   $y_1, y_2$: $2 + 3 = 5$
*   $y_2, y_1$: $2 + 3 = 5$
*   $y_1, x_1$: $2 + 4 = 6$
*   $y_2, x_1$: $2 + 4 = 6$
*   $x_1, y_1$: $2 + 4 = 6$
*   $x_1, y_2$: $2 + 4 = 6$

The sum of these is $34$.

Input #2

30 40 50

Output #2

616365902

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

API Response (JSON)

{
  "problem": {
    "name": "Paper Cutting",
    "description": {
      "content": "There is a rectangular sheet of paper with height $H+1$ and width $W+1$. We introduce an $xy$\\-coordinate system so that the four corners of the sheet are $(0, 0)$, $(W + 1, 0)$, $(0, H + 1)$ and $(W ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "keyence2019_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a rectangular sheet of paper with height $H+1$ and width $W+1$. We introduce an $xy$\\-coordinate system so that the four corners of the sheet are $(0, 0)$, $(W + 1, 0)$, $(0, H + 1)$ and $(W ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments