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.Linq;
using System.Collections.Generic;

namespace Contest
{
    class Program
    {
        static void Main(string[] args)
        {
            ABC145.C();
        }



        class ABC145
        {
            public static void A()
            {
                var R = Read.Int();
                Console.WriteLine(R * R);
            }
            public static void B()
            {
                var N = Read.Int();
                var S = Read.Str();
                Console.WriteLine(N % 2 == 0 && S.Substring(0, N / 2) == S.Substring(N / 2, N / 2) ? "Yes" : "No");
            }
            public static void C()
            {
                var N = Read.Int();
                var XY = Read.TupleInt(N);
                double sum = 0;
                for(int i = 0; i < N; ++i)
                {
                    var x1 = XY[i].Item1;
                    var y1 = XY[i].Item2;
                    for (int j = i + 1; j < N; ++j)
                    {
                        var x2 = XY[j].Item1;
                        var y2 = XY[j].Item2;
                        sum += Math.Sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2));
                    }
                }
                Console.WriteLine(sum / N * 2);
            }
            public static void D()
            {
                var X = Read.Int();
                var Y = Read.Int();
                if ((X + Y) % 3 != 0) { Console.WriteLine(0);return; }
                var a = X - (X + Y) / 3;
                var b = Y - (X + Y) / 3;
                Console.WriteLine(Util.Combin(a + b, 1000000007)(a + b, a));
            }
            public static void E()
            {
                var N = Read.Int();
                var T = Read.Int();
                var AB = Read.TupleInt(N);
                AB = AB.OrderBy(x => x.Item1).ToArray();
                var dp = new int[N + 1, T];
                for(int n = 0; n < N; ++n)
                {
                    var A = AB[n].Item1;
                    var B = AB[n].Item2;
                    for(int t = 0; t < T; ++t)
                    {
                        if (t >= A)
                        {
                            dp[n + 1, t] = Math.Max(dp[n, t], dp[n, t - A] + B);
                        }
                        else
                        {
                            dp[n + 1, t] = dp[n, t];
                        }
                    }
                }
                int max = 0;
                int maxB = 0;
                for(int n = N; n > 0; --n)
                {
                    max = Math.Max(max, dp[n, T - 1] + maxB);
                    maxB = Math.Max(maxB, AB[n - 1].Item2);
                }
                Console.WriteLine(max);
            }
            public static void F()
            {
                var N = Read.Int();
                var K = Read.Int();
                var H = Read.LongN();
                H = new long[] { 0 }.Concat(H).ToArray();
                var dp = new long[N + 1, N - K + 1];
                for (int len = 0; len <= N - K; ++len)
                {
                    dp[0, len] = len == 0 ? 0 : (long)1e18;
                }
                for (int n = 1; n <= N; ++n)
                {
                    dp[n, 0] = (long)1e18;
                    var h = H[n];
                    for(int len = 1; len <= N - K; ++len)
                    {
                        long min = (long)1e18;
                        for(int prev = 0; prev < n; ++prev)
                        {
                            min = Math.Min(min, dp[prev, len - 1] + Math.Max(0, h - H[prev]));
                        }
                        dp[n, len] = min;
                    }
                }
                Console.WriteLine(Enumerable.Range(1, N).Select(n => dp[n, N - K]).Min());
            }
        }


        class Util
        {
            public static long Inverse(long a, long p)
            {
                long b = p, c = 1, d = 0, e = 0, f = 0;
                while (b != 0)
                {
                    e = a / b;
                    f = b; b = a - e * b; a = f;
                    f = d; d = c - e * d; c = f;
                }
                c %= p;
                return c + (c < 0 ? p : 0);
            }

            public static long[] Facts(int n, long p)
            {
                var facts = new long[n + 1];
                facts[0] = 1;
                for (int i = 1; i <= n; ++i)
                {
                    facts[i] = (i * facts[i - 1]) % p;
                }
                return facts;
            }
            public static Func<int, int, long> Combin(int n, long P)
            {
                var fact = Facts(n, P);
                var inv = fact.Select(x => Inverse(x, P)).ToArray();
                return (N, M) =>
                {
                    if (N < 0 || N < M || M < 0) { return 0; }
                    return (((fact[N] * inv[M]) % P) * inv[N - M]) % P;
                };
            }
        }


        static class Read
        {
            static Queue<string> queue = new Queue<string>();
            static void Enqueue() { var ss = Console.ReadLine().Split(' '); foreach (var s in ss) { queue.Enqueue(s); } }
            public static int Int() { if (queue.Count == 0) { Enqueue(); } return Convert.ToInt32(queue.Dequeue()); }
            public static long Long() { if (queue.Count == 0) { Enqueue(); } return Convert.ToInt64(queue.Dequeue()); }
            public static string Str() { if (queue.Count == 0) { Enqueue(); } return queue.Dequeue(); }
            public static int[] IntN() { return Console.ReadLine().Split(' ').Select(s => Convert.ToInt32(s)).ToArray(); }
            public static long[] LongN() { return Console.ReadLine().Split(' ').Select(s => Convert.ToInt64(s)).ToArray(); }
            public static string[] StrN() { return Console.ReadLine().Split(' '); }
            public static Tuple<int, int>[] TupleInt(int n) { List<Tuple<int, int>> list = new List<Tuple<int, int>>(); for (int i = 0; i < n; ++i) { var ab = Read.IntN(); list.Add(new Tuple<int, int>(ab[0], ab[1])); } return list.ToArray(); }
            public static Tuple<long, long>[] TupleLong(int n) { List<Tuple<long, long>> list = new List<Tuple<long, long>>(); for (int i = 0; i < n; ++i) { var ab = Read.LongN(); list.Add(new Tuple<long, long>(ab[0], ab[1])); } return list.ToArray(); }
            public static Tuple<string, string>[] TupleStr(int n) { List<Tuple<string, string>> list = new List<Tuple<string, string>>(); for (int i = 0; i < n; ++i) { var ab = Read.StrN(); list.Add(new Tuple<string, string>(ab[0], ab[1])); } return list.ToArray(); }
            public static Tuple<double, double>[] TupleDouble(int n) { List<Tuple<double, double>> list = new List<Tuple<double, double>>(); for (int i = 0; i < n; ++i) { var ab = Read.LongN(); list.Add(new Tuple<double, double>(ab[0], ab[1])); } return list.ToArray(); }
            public static long[,] LongMatrix(int r, int c)
            {
                var mat = new long[r, c];
                for (int i = 0; i < r; ++i)
                {
                    var x = Read.LongN();
                    for (int j = 0; j < c; ++j)
                    {
                        mat[i, j] = x[j];
                    }
                }
                return mat;
            }
            public static Dictionary<int, HashSet<int>> AdjacencyList(int N, int M, bool isDirectional = false)
            {
                Dictionary<int, HashSet<int>> dict = new Dictionary<int, HashSet<int>>();
                for (int i = 0; i <= N; ++i) { dict.Add(i, new HashSet<int>()); }
                for (int i = 0; i < M; ++i)
                {
                    var ab = Read.IntN(); dict[ab[0]].Add(ab[1]);
                    if (!isDirectional) { dict[ab[1]].Add(ab[0]); }
                }
                return dict;
            }
            public static Dictionary<int, Dictionary<int, long>> AdjacencyListWithWeight(int N, int M)
            {
                Dictionary<int, Dictionary<int, long>> dict = new Dictionary<int, Dictionary<int, long>>();
                for (int i = 0; i <= N; ++i)
                {
                    dict.Add(i, new Dictionary<int, long>());
                }
                for (int i = 0; i < M; ++i)
                {
                    var ab = Read.IntN();
                    dict[ab[0]].Add(ab[1], ab[2]);
                    dict[ab[1]].Add(ab[0], ab[2]);
                }
                return dict;
            }
        }
    }
}