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 System.Text;
using System.Threading.Tasks;

namespace ConsoleApp1
{
    class CAverage
    {
        static void Main(string[] args)
        {
            var n = int.Parse(Console.ReadLine());

            var points = new List<Point>();

            for (int i = 0; i < n; i++)
            {
                var s = Console.ReadLine().Split(' ');
                points.Add(new Point(int.Parse(s[0]), int.Parse(s[1])));
            }

            List<double> allDistances = CreateAllDistances(points);

            // 2点の場合
            // ・2通り(2!)の経路
            // ・移動回数は1(2-1)回
            // ・移動パターンの全出現回数は2(2*1)回
            // ・移動パターンの各出現回数は1(2/2)回

            // 3点の場合
            // ・6(3!)通りの経路
            // ・移動回数は2(3-1)回
            // ・移動パターンの全出現回数は12(6*2)回
            // ・移動パターンの各出現回数は4(12/3)回

            int rootPattern = Factorial(points.Count);
            int moveCount = n - 1;
            int movePatternAllCount = rootPattern * moveCount;

            Console.WriteLine(CalcAvg(allDistances, rootPattern, movePatternAllCount));            
            //Console.ReadLine();
        }

        static List<double> CreateAllDistances(List<Point> points)
        {
            var allDistance = new List<double>();

            for (int i = 0; i < points.Count; i++)
            {
                for (int j = 0; j < points.Count; j++)
                {
                    if (j <= i)
                    {
                        continue;
                    }

                    allDistance.Add(CalcDistance(points[i], points[j]));
                }
            }

            return allDistance;
        }

        static double CalcAvg(List<double> allDistances, int rootPattern, int movePatternAllCount)
        {
            double distance = 0;
            int addCount = 0;
            while (addCount < movePatternAllCount)
            {
                foreach (var item in allDistances)
                {
                    distance += item;
                    addCount++;
                }
            }
            return distance / rootPattern;
        }

        static int Factorial(int n)
        {
            if (n == 0)
                return 1;
            return n * Factorial(n - 1);
        }

        static double CalcDistance(Point p1, Point p2)
        {
            return Math.Sqrt(Math.Pow(p1.X - p2.X, 2) + Math.Pow(p1.Y - p2.Y, 2));
        }
    }

    public class Point
    {
        public int X;
        public int Y;
        public Point(int x, int y)
        {
            this.X = x;
            this.Y = y;
        }
    }
}