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.ComponentModel.Design;
using System.Diagnostics;
using System.Linq;
using System.Runtime.Remoting.Proxies;
using System.Security.Policy;
using static System.Console;
 
 public static class Program
 {
     private static readonly long INF = long.MaxValue / 2;
     private static readonly int MOD = 1000000007;
 
     struct Pair<T, U> : IComparable<Pair<T, U>>
         where T : IComparable<T>
         where U : IComparable<U>
     {
         public T first;
         public U second;
         private bool CompWithFirst;
 
         public Pair(T f, U s, int comp)
         {
             first = f;
             second = s;
             CompWithFirst = comp == 1;
         }
 
         public int CompareTo(Pair<T, U> other)
         {
             return CompWithFirst ? first.CompareTo(other.first) : second.CompareTo(other.second);
         }
     }
 
     public class PriorityQueue<T> where T : IComparable<T>
     {
 
         private List<T> Q;
         private readonly int r;
 
         public PriorityQueue(bool ascending)
         {
             if (ascending)
             {
                 r = 1;
             }
             else
             {
                 r = -1;
             }
 
             Q = new List<T>();
         }
 
 
         private void PushHeap(List<T> list, T item)
         {
             int n = list.Count();
             list.Add(item);
 
             while (n != 0)
             {
                 int pIndex = (n - 1) / 2;
 
                 if (list[n].CompareTo(list[pIndex]) * r > 0)
                 {
                     Swap(Q, n, pIndex);
                 }
                 else
                 {
                     break;
                 }
 
                 n = pIndex;
             }
         }
 
         private void PopHeap(List<T> list)
         {
             int n = list.Count() - 1;
             list[0] = list[n];
             list.RemoveAt(n);
 
             int cur = 0;
             int comp;
 
             while (2 * cur + 1 <= n - 1)
             {
                 int c1 = 2 * cur + 1;
                 int c2 = 2 * (cur + 1);
                 if (c1 == n - 1)
                 {
                     comp = c1;
                 }
                 else
                 {
 
                     comp = list[c1].CompareTo(list[c2]) * r > 0 ? c1 : c2;
                 }
 
                 if (list[cur].CompareTo(list[comp]) * r < 0)
                 {
                     Swap(Q, cur, comp);
                 }
                 else
                 {
                     break;
                 }
 
                 cur = comp;
             }
         }
 
         private void Swap(List<T> list, int a, int b)
         {
             T keep = list[a];
             list[a] = list[b];
             list[b] = keep;
         }
 
 
 
         public void Enqueue(T value)
         {
             PushHeap(Q, value);
         }
 
         public T Dequeue()
         {
             T ret = Q[0];
             PopHeap(Q);
             return ret;
         }
 
         public T Peek()
         {
             return Q[0];
         }
 
         public int CountItem()
         {
             return Q.Count();
         }
 
         public bool IsEmpty()
         {
             return !Q.Any();
         }
     }
 
     static long ModInv(long a, long m)
     {
         long b = m, u = 1, v = 0;
         while (b != 0)
         {
             long t = a / b;
             a -= t * b;
             long tmp = a;
             a = b;
             b = tmp;
             u -= t * v;
             tmp = u;
             u = v;
             v = tmp;
         }
 
         u %= m;
         if (u < 0) u += m;
         return u;
     }
 
     public static long CalcComb(long n, long m)
     {
         //条件チェック
         if (n < m) return 0;
         if (n < 0 || m < 0) return 0;
 
         //nＣm ＝ nＣm-1 ×（n-m+1）/ m 
 //         return CalcFact(n) / (CalcFact(m) * CalcFact(n - m));
         //MODで割ってbの逆元を求める
         return CalcFact(n) * ModInv((CalcFact(m) * CalcFact(n - m)) % MOD, MOD) % MOD;
     }
 
     /// <summary>
     /// 階乗を計算する。
     /// </summary>
     /// <param name="n">n</param>
     /// <returns>nの階乗（n!）</returns>
     public static long CalcFact(long n)
     {
         if (n <= 0)
         {
             return 1;
         }
 
         return n * CalcFact(n - 1) % MOD;
     }


     partial struct Pos
     {
         public int x;
         public int y;
 
         public Pos(int x, int y)
         {
             this.x = x;
             this.y = y;
         }
         
         public static Pos operator +(Pos p1, Pos p2)
         {
             var result = new Pos();
             result.x = p2.x + p1.x;
             result.y = p2.y + p1.y;
             return result;
         }
 
         public static Pos operator -(Pos p1, Pos p2)
         {
             var result = new Pos();
             result.x = p2.x - p1.x;
             result.y = p2.y - p1.y;
             return result;
         }
     }
     
     class Modular
     {
         private const int M = 1000000007;
         private long value;
 
         public Modular(long value)
         {
             this.value = value;
         }
 
         public static implicit operator Modular(long a)
         {
             var m = a % M;
             return new Modular((m < 0) ? m + M : m);
         }
 
         public static Modular operator +(Modular a, Modular b)
         {
             return a.value + b.value;
         }
 
         public static Modular operator -(Modular a, Modular b)
         {
             return a.value - b.value;
         }
 
         public static Modular operator *(Modular a, Modular b)
         {
             return a.value * b.value;
         }
 
         private static Modular Pow(Modular a, int n)
         {
             switch (n)
             {
                 case 0:
                     return 1;
                 case 1:
                     return a;
                 default:
                     var p = Pow(a, n / 2);
                     return p * p * Pow(a, n % 2);
             }
         }
 
         public static Modular operator /(Modular a, Modular b)
         {
             return a * Pow(b, M - 2);
         }
 
         private static readonly List<int> facs = new List<int> {1};
 
         private static Modular Fac(int n)
         {
             for (int i = facs.Count; i <= n; ++i)
             {
                 facs.Add((int) (Math.BigMul(facs.Last(), i) % M));
             }
 
             return facs[n];
         }
 
         public static Modular Ncr(int n, int r)
         {
             return (n < r) ? 0
                 : (n == r) ? 1
                 : Fac(n) / (Fac(r) * Fac(n - r));
         }
 
         public static explicit operator int(Modular a)
         {
             return (int) a.value;
         }
     }
     
     static class Permutation<T>
     {
         private static void Search(List<T[]> perms, Stack<T> stack, T[] a)
         {
             int N = a.Length;
             if (N == 0)
             {
                 perms.Add(stack.Reverse().ToArray());
             }
             else
             {
                 var b = new T[N - 1];
                 Array.Copy(a, 1, b, 0, N - 1);
                 for (int i = 0; i < a.Length; ++i)
                 {
                     stack.Push(a[i]);
                     Search(perms, stack, b);
                     if (i < b.Length) { b[i] = a[i]; }
                     stack.Pop();
                 }
             }
         }
         public static IEnumerable<T[]> All(IEnumerable<T> src)
         {
             var perms = new List<T[]>();
             Search(perms, new Stack<T>(), src.ToArray());
             return perms;
         }
     }
 
     public static int ReadLineToInt()
     {
         return int.Parse(ReadLine());
     }
     
     public static long ReadLineToLong()
     {
         return long.Parse(ReadLine());
     }
     
     public static int[] ReadLineToInts()
     {
         return ReadLine().Split(' ').Select(int.Parse).ToArray();
     }
     
     public static long[] ReadLineToLongs()
     {
         return ReadLine().Split(' ').Select(long.Parse).ToArray();
     }
     
     static int nibunn(int[] array, int target)
     {
         int L = 0;
         int R = array.Length; 
             
         while(R - L > 1)
         {
             int mid = (R + L) / 2;
             if (array[mid] > target) L = mid;
             else R = mid;
         }
         return L;
     }

     struct P
     {
         public int first;
         public int second;

         public P(int a, int b)
         {
             first = a;
             second = b;
         }
     }
    
     static long GCD(long x, long y)
     {
         if (y == 0) return x;
         return GCD(y, x%y);
     }

     static P f(int x)
     {
         int a = x % 10;
         int b = 0;
         while (x != 0)
         {
             b = x;
             x /= 10;
         }
         return new P(a,b);
     }

//     static void dfs(int v, int p = -1)
//     {
//         
//         foreach (var e in Tree[0])
//         {
//             dfs(e, v);
//         }
//     }
//     static List<int>[] Tree;
//     private static int[,] Edge;
//
//     static void SetColor(int v1, int v2, int c)
//     {
//         if (v1 < v2)
//         {
//             Edge[v1, v2] = c;
//         }
//         else
//         {
//             Edge[v2, v1] = c;
//         }
//     }
//     public static void Main(string[] args)
//     {
//         int n = ReadLineToInt();
//         Tree = new List<int>[n];
//         Edge = new int[n-1,n-1];
//         for (int i = 0; i < n; i++)
//         {
//             Tree[i] = new List<int>();
//         }
//         
//         for (int i = 0; i < n-1; i++)
//         {
//             for (int j = 0; j < n - 1; j++)
//             {
//                 Edge[i, j] = INF;
//             }
//         }
//
//         
//         for (int i = 0; i < n-1; i++)
//         {
//             var l1 = ReadLineToInts();
//             int a = l1[0];
//             int b = l1[1];
//             a--;
//             b--;
//             Tree[a].Add(b);
//             Tree[b].Add(a);
//         }
//         
//         dfs(0);
//     }

     private static List<Pos> PosList;
        
     static double dist(int i, int j)
     {
         var pos = PosList[j-1] - PosList[i-1];
         return Math.Sqrt(Math.Pow(pos.x, 2) + Math.Pow(pos.y, 2));
     }

     public static void Main(string[] args)
     {
         int n = ReadLineToInt();
         PosList = new List<Pos>();
         foreach (int i in Enumerable.Range(0, n))
         {
             var l = ReadLineToInts();
             var pos = new Pos(x: l[0], y: l[1]);
             PosList.Add(pos);
         }
         
         double sum = 0;
         var permutation = Permutation<int>.All(Enumerable.Range(1,n));
         foreach (var p in permutation) {
             for (int i = 0; i < n-1; i++)
             {
                 sum += dist(p[i], p[i+1]);
             }
         }
         double frac = 1;
         for (int i = 2; i <= n; i++)
         {
             frac *= i;
         }
         WriteLine($"{sum/frac:F10}");
         
     }
 }