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.Linq;
using static Input;
using NX = System.Int64;


public class Prog
{
    public static void Solve()
    {
        int N = NextInt();

        NextPermutation NP = new NextPermutation();
        var np = NP.Run(N);
        List<Tuple<int, int>> xy = new List<Tuple<int, int>>();
        for (int i = 0; i < N; i++)
        {
            int x = NextInt();
            int y = NextInt();
            xy.Add(new Tuple<int, int>(x, y));
        }
        double ans = 0;
        foreach (var l in np)
        {
            double calc = 0;
            bool first = true;
            Tuple<int, int> bef = new Tuple<int, int>(0, 0); ;
            foreach (var i in l)
            {
                if (first) { bef = xy[i]; first = false; continue; }
                calc += Math.Pow((xy[i].Item1 - bef.Item1), 2);
                calc += Math.Pow((xy[i].Item2 - bef.Item2), 2);
                bef = xy[i];
                ans += Math.Sqrt(calc);
                calc = 0;
            }
        }
        for (int i = 1; i <= N; i++) ans /= i;
        Console.WriteLine(ans);
    }
    public static void Main()
    {
        var sw = new System.IO.StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        Console.SetOut(sw);
        Solve();
        Console.Out.Flush();
        return;
    }

}

public class NextPermutation
{
    public List<List<int>> ret;
    private Queue<List<int>> Q;

    public NextPermutation()
    {
        ret = new List<List<int>>();
        Q = new Queue<List<int>>();
    }

    public List<List<int>> Run(int N)
    {
        for (int i = 0; i < N; i++)
        {
            List<int> box = new List<int>();
            box.Add(i);
            Q.Enqueue(box);
        }
        while (Q.Count() != 0)
        {
            List<int> box = Q.Dequeue();
            if (box.Count() == N) { ret.Add(box); }
            else
            {
                for (int i = 0; i < N; i++)
                {
                    if (!box.Contains(i))
                    {
                        List<int> b = new List<int>(box);
                        b.Add(i);
                        Q.Enqueue(b);
                    }
                }
            }
        }
        return ret;
    }
}

public class Input
{
    public const long MOD = 1000000007;
    public const int INF = 1000000007;
    private static Queue<string> q = new Queue<string>();
    private static void Confirm() { if (q.Count == 0) foreach (var s in Console.ReadLine().Split(' ')) q.Enqueue(s); }
    public static int NextInt() { Confirm(); return int.Parse(q.Dequeue()); }
    public static long NextLong() { Confirm(); return long.Parse(q.Dequeue()); }
    public static string NextString() { Confirm(); return q.Dequeue(); }
    public static double NextDouble() { Confirm(); return double.Parse(q.Dequeue()); }
    public static int[] LineInt() { return Console.ReadLine().Split(' ').Select(int.Parse).ToArray(); }
    public static long[] LineLong() { return Console.ReadLine().Split(' ').Select(long.Parse).ToArray(); }
    public static string[] LineString() { return Console.ReadLine().Split(' ').ToArray(); }
    public static double[] LineDouble() { return Console.ReadLine().Split(' ').Select(double.Parse).ToArray(); }
    public long Abs(long a, long b) => Math.Abs(a - b);
}