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.Text;
using System.Linq;
using System.Collections;
using System.Collections.Generic;
using static System.Console;
using static System.Math;

namespace AtCoder
{
    public class Program
    {
        public static void Main(string[] args)
        {
            new Program().Solve(new ConsoleInput(Console.In, ' '));
        }

        public void Solve(ConsoleInput cin)
        {
            int N = cin.ReadInt;
            var x = new List<int>();
            var y = new List<int>();
            var seq = new List<int>();

            for (int i=0; i<N; i++)
            {
                x.Add(cin.ReadInt);
                y.Add(cin.ReadInt);
                seq.Add(i);
            }

            double ans = 0;
            int perms = 0;
            foreach (var permu in Permutate(seq, seq.Count))
            {
                perms++;
                for (int i=0; i<permu.Count-1; i++)
                {
                    ans += Sqrt(Pow((x[int.Parse(permu[i].ToString())]-x[int.Parse(permu[i+1].ToString())]),2) + Pow((y[int.Parse(permu[i].ToString())] - y[int.Parse(permu[i + 1].ToString())]), 2));
                }                    
            }
            WriteLine(ans / perms);
        }

        static long GcdArray(long[] array)
        {
            var ans = array[0];
            for(var i=1; i<array.Length; i++)
            {
                ans = Gcd(ans, array[i]);
            }
            return ans;
        }

        static long Gcd(long a, long b)
        {
            var v = new[] { a, b };
            while (v[1] != 0) { v = new[] { v[1], v[0] % v[1] }; }
            return v[0];
        }

        static List<long> GetDivisor(long n)
        {
            List<long> divisor = new List<long>();
            for (long i=1; i*i<=n; i++)
            {
                if (n % i == 0)
                {
                    divisor.Add(i);
                    if (i != n / i)
                        divisor.Add(n / i);
                }
            }
            divisor.Sort();
            return divisor;

        }

        // 互いに素
        static bool coprime(long a, long b)
        {
            return (Gcd(a, b) == 1);
        }

        static long Lcm(long a, long b)
        {
            return a / Gcd(a, b) * b;
        }

        static bool ChMin<Type>(ref Type a, Type b)
            where Type : IComparable
        {   
            if(a.CompareTo(b) > 0)
            {
                a = b;
                return true;
            }
            return false;
        }

        static bool ChMax<Type>(ref Type a, Type b)
            where Type : IComparable
        {
            if (a.CompareTo(b) < 0)
            {
                a = b;
                return true;
            }
            return false;
        }

        public static void RotateRight(IList sequence, int count)
        {
            object tmp = sequence[count - 1];
            sequence.RemoveAt(count - 1);
            sequence.Insert(0, tmp);
        }

        public static IEnumerable<IList> Permutate(IList sequence, int count)
        {
            if (count == 1) yield return sequence;
            else
            {
                for (int i = 0; i < count; i++)
                {
                    foreach (var perm in Permutate(sequence, count - 1))
                        yield return perm;
                    RotateRight(sequence, count);
                }
            }
        }
    }

    public class ConsoleInput
    {
        private readonly System.IO.TextReader _stream;
        private char _separator = ' ';
        private Queue<string> inputStream;
        public ConsoleInput(System.IO.TextReader stream, char separator = ' ')
        {
            this._separator = separator;
            this._stream = stream;
            inputStream = new Queue<string>();
        }
        public string Read
        {
            get
            {
                if (inputStream.Count != 0) return inputStream.Dequeue();
                string[] tmp = _stream.ReadLine().Split(_separator);
                for (int i = 0; i < tmp.Length; ++i)
                    inputStream.Enqueue(tmp[i]);
                return inputStream.Dequeue();
            }
        }
        public string ReadLine { get { return _stream.ReadLine(); } }
        public int ReadInt { get { return int.Parse(Read); } }
        public long ReadLong { get { return long.Parse(Read); } }
        public double ReadDouble { get { return double.Parse(Read); } }
        public string[] ReadStrArray(long N) { var ret = new string[N]; for (long i = 0; i < N; ++i) ret[i] = Read; return ret; }
        public int[] ReadIntArray(long N) { var ret = new int[N]; for (long i = 0; i < N; ++i) ret[i] = ReadInt; return ret; }
        public long[] ReadLongArray(long N) { var ret = new long[N]; for (long i = 0; i < N; ++i) ret[i] = ReadLong; return ret; }
    }

    // From http://visualstudiomagazine.com/articles/2012/11/01/priority-queues-with-c.aspx
    public class PriorityQueue<T> where T : IComparable<T>
    {
        private List<T> data;

        public PriorityQueue()
        {
            this.data = new List<T>();
        }

        public void Enqueue(T item)
        {
            data.Add(item);
            int ci = data.Count - 1; // child index; start at end
            while (ci > 0)
            {
                int pi = (ci - 1) / 2; // parent index
                if (data[ci].CompareTo(data[pi]) >= 0)
                    break; // child item is larger than (or equal) parent so we're done
                T tmp = data[ci];
                data[ci] = data[pi];
                data[pi] = tmp;
                ci = pi;
            }
        }

        public T Dequeue()
        {
            // assumes pq is not empty; up to calling code
            int li = data.Count - 1; // last index (before removal)
            T frontItem = data[0];   // fetch the front
            data[0] = data[li];
            data.RemoveAt(li);

            --li; // last index (after removal)
            int pi = 0; // parent index. start at front of pq
            while (true)
            {
                int ci = pi * 2 + 1; // left child index of parent
                if (ci > li)
                    break;  // no children so done
                int rc = ci + 1;     // right child
                if (rc <= li && data[rc].CompareTo(data[ci]) < 0) // if there is a rc (ci + 1), and it is smaller than left child, use the rc instead
                    ci = rc;
                if (data[pi].CompareTo(data[ci]) <= 0)
                    break; // parent is smaller than (or equal to) smallest child so done
                T tmp = data[pi];
                data[pi] = data[ci];
                data[ci] = tmp; // swap parent and child
                pi = ci;
            }
            return frontItem;
        }

        public T Peek()
        {
            T frontItem = data[0];
            return frontItem;
        }

        public int Count()
        {
            return data.Count;
        }

        public override string ToString()
        {
            string s = "";
            for (int i = 0; i < data.Count; ++i)
                s += data[i].ToString() + " ";
            s += "count = " + data.Count;
            return s;
        }

        public bool IsConsistent()
        {
            // is the heap property true for all data?
            if (data.Count == 0)
                return true;
            int li = data.Count - 1; // last index
            for (int pi = 0; pi < data.Count; ++pi)
            { // each parent index
                int lci = 2 * pi + 1; // left child index
                int rci = 2 * pi + 2; // right child index

                if (lci <= li && data[pi].CompareTo(data[lci]) > 0)
                    return false; // if lc exists and it's greater than parent then bad.
                if (rci <= li && data[pi].CompareTo(data[rci]) > 0)
                    return false; // check the right child too.
            }
            return true; // passed all checks
        }
        // IsConsistent
    }
}