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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static Asakatsu20200317B.Input;
using static System.Math;

namespace Asakatsu20200317B
{
    class Input
    {
        private static String String() => Console.ReadLine();

        private static IEnumerable<string> Split() => String().Split(' ');

        private static long[] Long() => Split().Select(long.Parse).ToArray();

        /// <summary>
        /// 1行の文字列の入力
        /// </summary>
        public static void @in(out string s) => s = String();

        /// <summary>
        /// 複数行の文字列の入力
        /// </summary>
        public static void @in(long rowNumber, out string[] s)
        {
            s = new String[rowNumber];
            for (long i = 0; i < rowNumber; i++) @in(out s[i]);
        }

        /// <summary>
        /// 1行に書かれた1つの整数の入力
        /// </summary>
        public static void @in(out int i)
        {
            i = int.Parse(String());
        }

        /// <summary>
        /// 1行に書かれた1つの整数の入力
        /// </summary>
        public static void @in(out long a) => a = Long()[0];

        /// <summary>
        /// 1行に書かれた2つの整数の入力
        /// </summary>
        public static void @in(out long a, out long b)
        {
            long[] lArr = Long();
            a = lArr[0];
            b = lArr[1];
        }

        /// <summary>
        /// 1行に書かれた3つの整数の入力
        /// </summary>
        public static void @in(out long a, out long b, out long c)
        {
            long[] lArr = Long();
            a = lArr[0];
            b = lArr[1];
            c = lArr[2];
        }

        /// <summary>
        /// 1行に書かれた4つの整数の入力
        /// </summary>
        public static void @in(out long a, out long b, out long c, out long d)
        {
            long[] lArr = Long();
            a = lArr[0];
            b = lArr[1];
            c = lArr[2];
            d = lArr[3];
        }

        /// <summary>
        /// 1行に書かれた5つの整数の入力
        /// </summary>
        public static void @in(out long a, out long b, out long c, out long d, out long e)
        {
            long[] lArr = Long();
            a = lArr[0];
            b = lArr[1];
            c = lArr[2];
            d = lArr[3];
            e = lArr[4];
        }

        /// <summary>
        /// 1行に書かれた複数の整数列の入力
        /// </summary>
        public static void @in(out long[] lArr) => lArr = Long();

        /// <summary>
        /// rowNumber行に書かれた1つの整数列の入力
        /// </summary>
        public static void @in(long rowNumber, out long[] lArr)
        {
            lArr = new long[rowNumber];
            for (long i = 0; i < rowNumber; i++) @in(out lArr[i]);
        }

        /// <summary>
        /// rowNumber行に書かれた2つの整数列の入力
        /// </summary>
        public static void @in(long rowNumber, out long[] a, out long[] b)
        {
            a = new long[rowNumber];
            b = new long[rowNumber];
            for (int i = 0; i < rowNumber; i++) @in(out a[i], out b[i]);
        }

        /// <summary>
        /// rowNumber行に書かれた3つの整数列の入力
        /// </summary>
        public static void @in(long rowNumber, out long[] a, out long[] b, out long[] c)
        {
            a = new long[rowNumber];
            b = new long[rowNumber];
            c = new long[rowNumber];
            for (int i = 0; i < rowNumber; i++) @in(out a[i], out b[i], out c[i]);
        }

        /// <summary>
        /// rowNumber行に書かれた4つの整数列の入力
        /// </summary>
        public static void @in(long rowNumber, out long[] a, out long[] b, out long[] c, out long[] d)
        {
            a = new long[rowNumber];
            b = new long[rowNumber];
            c = new long[rowNumber];
            d = new long[rowNumber];
            for (int i = 0; i < rowNumber; i++) @in(out a[i], out b[i], out c[i], out d[i]);
        }

        /// <summary>
        /// h行w列の整数の行列の入力
        /// </summary>
        public static void @in(long h, long w, out long[][] a)
        {
            a = new long[h][];
            for (long i = 0; i < h; i++) @in(out a[i]);
        }

        /// <summary>
        /// 1行に書かれた1つの小数の入力
        /// </summary>
        public static void @in(out double d) => d = double.Parse(String());

        /// <summary>
        /// 1行に書かれた小数の配列の入力
        /// </summary>
        public static void @in(out double[] dArr) => dArr = Split().Select(double.Parse).ToArray();
    }

    class Program
    {
        public static void Main(string[] args)
        {
            StreamWriter streamWriter = new StreamWriter(Console.OpenStandardOutput()) {AutoFlush = false};
            Console.SetOut(streamWriter);
            new Solver().Solve();
            Console.Out.Flush();
        }

        public static void Debug()
        {
            new Solver().Solve();
        }
    }

    class Solver
    {
        private int N;
        private long[] x;
        private long[] y;

        public void Solve()
        {
            @in(out N);
            @in(N, out x, out y);

            var p = new Permutation<int>(Enumerable.Range(0, N).ToList());

            double sum = 0;
            for (int i = 0; i < p.permutation.Count; i++)
            {
                var c = p.permutation[i];
                double d = 0;
                for (int j = 1; j < c.Count; j++)
                {
                    d += Sqrt((x[c[j - 1]] - x[c[j]]) * (x[c[j - 1]] - x[c[j]]) +
                              (y[c[j - 1]] - y[c[j]]) * (y[c[j - 1]] - y[c[j]]));
                }

                sum += d;
            }

            Console.WriteLine(sum / p.permutation.Count);
        }
    }

    /// <summary>
    /// 順列を生成する
    /// </summary>
    /// <typeparam name="T"></typeparam>
    public class Permutation<T>
    {
        public Permutation(List<T> list)
        {
            Permutate(list);
        }

        public List<List<T>> permutation = new List<List<T>>();

        private void Permutate(List<T> rest, List<T> current = null)
        {
            if (current == null) current = new List<T>();

            foreach (var l in rest)
            {
                var next = new List<T>();
                foreach (T v in current)
                {
                    next.Add(v);
                }

                next.Add(l);

                if (rest.Count == 1)
                {
                    permutation.Add(next);
                    continue;
                }

                var nextRest = new List<T>();
                foreach (T r in rest)
                {
                    if (r.Equals(l)) continue;
                    nextRest.Add(r);
                }

                Permutate(nextRest, next);
            }
        }
    }
}