Average Length

AtCoder
IDabc145_c
Time2000ms
Memory256MB
Difficulty
There are $N$ towns in a coordinate plane. Town $i$ is located at coordinates ($x_i$, $y_i$). The distance between Town $i$ and Town $j$ is $\sqrt{\left(x_i-x_j\right)^2+\left(y_i-y_j\right)^2}$. There are $N!$ possible paths to visit all of these towns once. Let the length of a path be the distance covered when we start at the first town in the path, visit the second, third, $\dots$, towns, and arrive at the last town (assume that we travel in a straight line from a town to another). Compute the average length of these $N!$ paths. ## Constraints * $2 \leq N \leq 8$ * $-1000 \leq x_i \leq 1000$ * $-1000 \leq y_i \leq 1000$ * $\left(x_i, y_i\right) \neq \left(x_j, y_j\right)$ (if $i \neq j$) * (Added 21:12 JST) All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $x_1$ $y_1$ $:$ $x_N$ $y_N$ [samples]
Samples
Input #1
3
0 0
1 0
0 1
Output #1
2.2761423749

There are six paths to visit the towns: $1$ → $2$ → $3$, $1$ → $3$ → $2$, $2$ → $1$ → $3$, $2$ → $3$ → $1$, $3$ → $1$ → $2$, and $3$ → $2$ → $1$.
The length of the path $1$ → $2$ → $3$ is $\sqrt{\left(0-1\right)^2+\left(0-0\right)^2} + \sqrt{\left(1-0\right)^2+\left(0-1\right)^2} = 1+\sqrt{2}$.
By calculating the lengths of the other paths in this way, we see that the average length of all routes is:
$\frac{\left(1+\sqrt{2}\right)+\left(1+\sqrt{2}\right)+\left(2\right)+\left(1+\sqrt{2}\right)+\left(2\right)+\left(1+\sqrt{2}\right)}{6} = 2.276142...$
Input #2
2
-879 981
-866 890
Output #2
91.9238815543

There are two paths to visit the towns: $1$ → $2$ and $2$ → $1$. These paths have the same length.
Input #3
8
-406 10
512 859
494 362
-955 -475
128 553
-986 -885
763 77
449 310
Output #3
7641.9817824387
API Response (JSON)
{
  "problem": {
    "name": "Average Length",
    "description": {
      "content": "There are $N$ towns in a coordinate plane. Town $i$ is located at coordinates ($x_i$, $y_i$). The distance between Town $i$ and Town $j$ is $\\sqrt{\\left(x_i-x_j\\right)^2+\\left(y_i-y_j\\right)^2}$. Ther",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc145_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ towns in a coordinate plane. Town $i$ is located at coordinates ($x_i$, $y_i$). The distance between Town $i$ and Town $j$ is $\\sqrt{\\left(x_i-x_j\\right)^2+\\left(y_i-y_j\\right)^2}$.\nTher...",
      "is_translate": false,
      "language": "English"
    }
  ]
}
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