Problem:
Score : 300 points

Problem Statement

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

Output

Print the average length of the paths.
Your output will be judges as correct when the absolute difference from the judge's output is at most 10^{-6}.

Sample Input 1

3
0 0
1 0
0 1

Sample 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...

Sample Input 2

2
-879 981
-866 890

Sample Output 2

91.9238815543

There are two paths to visit the towns: 1 → 2 and 2 → 1. These paths have the same length.

Sample Input 3

8
-406 10
512 859
494 362
-955 -475
128 553
-986 -885
763 77
449 310

Sample Output 3

7641.9817824387

Solution in Rust (1.15.1) (judge verdict: Accepted)

//! https://github.com/hatoo/competitive-rust-snippets
//!
//! MIT License
//!
//! Copyright (c) 2018 hatoo
//!
//! Permission is hereby granted, free of charge, to any person obtaining a copy
//! of this software and associated documentation files (the "Software"), to deal
//! in the Software without restriction, including without limitation the rights
//! to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
//! copies of the Software, and to permit persons to whom the Software is
//! furnished to do so, subject to the following conditions:
//!
//! The above copyright notice and this permission notice shall be included in all
//! copies or substantial portions of the Software.
//!
//! THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
//! IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
//! FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
//! AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
//! LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
//! OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
//! SOFTWARE.
#![allow(unused_imports, unused_attributes, unused_macros, dead_code)]
use std::cmp::{max, min, Ordering};
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
use std::io::{stdin, stdout, BufWriter, Write};
use std::iter::FromIterator;
#[macro_export]
macro_rules !get {(@inner [$src :expr ] chars ) =>{{let mut buf =String ::new () ;$src .read_line (&mut buf ) .unwrap () ;buf .trim () .chars () .collect ::<Vec <char >>() } } ;(@inner [$src :expr ] usize1 ) =>{{get !(@inner [$src ] usize ) -1 } } ;(@inner [$src :expr ] [usize1 ] ) =>{{get !(@inner [$src ] [usize ] ) .into_iter () .map (|v |v -1 ) .collect ::<Vec <usize >>() } } ;(@inner [$src :expr ] [[usize1 ] ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] [usize1 ] ) ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [usize1 ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] [usize1 ] ) ) .flatten () .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [[chars ] ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] chars ) ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [chars ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] chars ) ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [($($tt :tt ) ,*) ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] ($($tt ) ,*) ) ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] ($($tt :tt ) ,*) ) =>{{let mut buf :String =String ::new () ;$src .read_line (&mut buf ) .unwrap () ;let mut iter =buf .split_whitespace () ;($(get !(@inner_elem_parse [$tt ] iter .next () .unwrap () ) ,) *) } } ;(@inner [$src :expr ] [$t :ty ] ) =>{{let mut buf =String ::new () ;$src .read_line (&mut buf ) .unwrap () ;buf .trim () .split_whitespace () .map (|t |t .parse ::<$t >() .unwrap () ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [[$t :ty ] ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] [$t ] ) ) .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] [$t :ty ;$n :expr ] ) =>{{(0 ..$n ) .map (|_ |get !(@inner [$src ] [$t ] ) ) .flatten () .collect ::<Vec <_ >>() } } ;(@inner [$src :expr ] $t :ty ) =>{{let mut buf =String ::new () ;$src .read_line (&mut buf ) .unwrap () ;buf .trim () .split_whitespace () .next () .unwrap () .parse ::<$t >() .unwrap () } } ;(@inner_elem_parse [usize1 ] $elem :expr ) =>{{get !(@inner_elem_parse [usize ] $elem ) -1 } } ;(@inner_elem_parse [$t :ty ] $elem :expr ) =>{{$elem .parse ::<$t >() .unwrap () } } ;($tt :tt ) =>{{use std ::io ::BufRead ;let get_stdin =std ::io ::stdin () ;let mut locked_stdin =get_stdin .lock () ;get !(@inner [&mut locked_stdin ] $tt ) } } ;}
macro_rules !debug {($($a :expr ) ,*) =>{eprintln !(concat !($(stringify !($a ) ," = {:?}, " ) ,*) ,$($a ) ,*) ;} }
macro_rules !echo {($($a :expr ) ,*) =>{let mut s =Vec ::new () ;$(s .push (format !("{}" ,$a ) ) ;) *println !("{}" ,s .join (" " ) ) ;} }
const BIG_STACK_SIZE: bool = true;
/// Ported from:
/// - [bluss/permutohedron](https://github.com/bluss/permutohedron)
/// - [hatoo/competitive-rust-snippets](https://github.com/hatoo/competitive-rust-snippets)
pub trait LexicalPermutation {
    /// Return  if the slice was permuted, Error: Command failed: false if it is already
    /// at the last ordered permutation.
    fn next_permutation(&mut self) -> bool;
    /// Return  if the slice was permuted, Error: Command failed: false if it is already
    /// at the first ordered permutation.
    fn prev_permutation(&mut self) -> bool;
}
impl<T> LexicalPermutation for [T]
where
    T: Ord,
{
    /// Original author in Rust: Thomas Backman <serenity@exscape.org>
    fn next_permutation(&mut self) -> bool {
        if self.len() < 2 {
            return false;
        }
        let mut i = self.len() - 1;
        while i > 0 && self[i - 1] >= self[i] {
            i -= 1;
        }
        if i == 0 {
            return false;
        }
        let mut j = self.len() - 1;
        while j >= i && self[j] <= self[i - 1] {
            j -= 1;
        }
        self.swap(j, i - 1);
        self[i..].reverse();
        true
    }
    fn prev_permutation(&mut self) -> bool {
        if self.len() < 2 {
            return false;
        }
        let mut i = self.len() - 1;
        while i > 0 && self[i - 1] <= self[i] {
            i -= 1;
        }
        if i == 0 {
            return false;
        }
        self[i..].reverse();
        let mut j = self.len() - 1;
        while j >= i && self[j - 1] < self[i - 1] {
            j -= 1;
        }
        self.swap(i - 1, j);
        true
    }
}
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}
fn solve() {
    let n = get!(usize);
    let xy = get!([(f64, f64); n]);
    let mut perm = (0..n).collect::<Vec<usize>>();
    let mut pat = 1;
    let mut sum_d = 0.0;

    for i in 0..(n - 1) {
        let (fx, fy) = xy[perm[i]];
        let (tx, ty) = xy[perm[i + 1]];
        sum_d += ((tx - fx).powi(2i32) + (ty - fy).powi(2i32)).sqrt();
    }

    while perm.next_permutation() {
        pat += 1;
        for i in 0..(n - 1) {
            let (fx, fy) = xy[perm[i]];
            let (tx, ty) = xy[perm[i + 1]];
            sum_d += ((tx - fx).powi(2i32) + (ty - fy).powi(2i32)).sqrt();
        }
    }

    echo!(sum_d / pat as f64);
}