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.Diagnostics;
using System.Globalization;
using System.IO;
using System.Linq;
using System.Runtime.CompilerServices;
using System.Text;

namespace AtCoder
{
    static class Program_D
    {
        class Solver
        {
            int ri => sc.Integer();
            long rl => sc.Long();
            double rd => sc.Double();
            string rs => sc.Scan();
            public StreamScanner sc = new StreamScanner(Console.OpenStandardInput());

            static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }

            private int[] pos;
            private int[] path;
            private int pathNum;
            private int n;
            double lenSum = 0;


            public void Solve()
            {
                n = ri;
                pos = new int[n*2];
                path = new int[n];
                pathNum = 1;
                for(int i = 0; i < n; ++i)
                {
                    pathNum *= i + 1;
                    pos[i * 2 + 0] = ri;
                    pos[i * 2 + 1] = ri;
                }
                

                var c = new int[n];
                for(int i=0;i<n;++i)
                {
                    c[i] = i;
                }
                Create(c, 0);
                Wl(lenSum);
            }

            private void Create(int[] c, int nth)
            {
                if (nth == n)
                {
                    Calc();
                    return;
                }
                for(int i=0;i<c.Length;++i)
                {
                    path[nth] = c[i];
                    var newPath = new int[c.Length - 1];
                    int dst = 0;
                    for(int j=0; j<c.Length;++j)
                    {
                        if (j == i) continue;
                        newPath[dst] = c[j];
                        dst++;
                    }
                    Create(newPath,nth+1);
                }
            }

            private void Calc()
            {
                double len = 0;
                for(int i=0;i<path.Length-1;++i)
                {
                    var s = path[i] * 2;
                    var d = path[i+1] * 2;
                    len += Math.Sqrt(
                        (pos[s]-pos[d])* (pos[s] - pos[d])+ (pos[s+1] - pos[d+1]) * (pos[s+1] - pos[d+1])
                        );
                }
                lenSum += len / pathNum;
            }





            // ----------------------------------------------------------------------------


            string Rl()
            {
                return Console.ReadLine();
            }
            void Wl(UInt64 val)
            {
                Console.WriteLine(val);
            }
            void Wl(Int64 val)
            {
                Console.WriteLine(val);
            }
            void Wl(uint val)
            {
                Console.WriteLine(val);
            }
            void Wl(int val)
            {
                Console.WriteLine(val);
            }
            void Wl(double val)
            {
                Console.WriteLine(val);
            }
            void Wl(string mes)
            {
                Console.WriteLine(mes);
            }
            public int Int()
            {
                return int.Parse(Console.ReadLine());
            }

            public int[] Ints(int subtract)
            {
                return Console.ReadLine().Split(' ').Select(s => int.Parse(s) - subtract).ToArray();
            }
            public Int64[] Int64s(Int64 subtract)
            {
                return Console.ReadLine().Split(' ').Select(s => Int64.Parse(s) - subtract).ToArray();
            }
            public UInt64[] UInt64s(UInt64 subtract)
            {
                return Console.ReadLine().Split(' ').Select(s => UInt64.Parse(s) - subtract).ToArray();
            }


        }



        static void Main(string[] args)
        {
            Console.SetOut(new Printer(Console.OpenStandardOutput()) { AutoFlush = false });
            new Solver().Solve();
            Console.Out.Flush();
        }
    }

    public class Printer : StreamWriter
    {
        public override IFormatProvider FormatProvider => CultureInfo.InvariantCulture;
        public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
    }

    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }

        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;

        private byte read()
        {
            if (isEof) return 0;
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0)
                {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }

        public char Char()
        {
            byte b = 0;
            do b = read(); while ((b < 33 || 126 < b) && !isEof);
            return (char)b;
        }
        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read()) sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n' && b != 0; b = (char)read()) if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long() { return isEof ? long.MinValue : long.Parse(Scan()); }
        public int Integer() { return isEof ? int.MinValue : int.Parse(Scan()); }
        public double Double() { return isEof ? double.NaN : double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }

    struct ModInt
    {
        const int MOD = 1000000007;
        long Data;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public ModInt(long data) { if ((Data = data % MOD) < 0) Data += MOD; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static implicit operator long(ModInt modInt) => modInt.Data;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static implicit operator ModInt(long val) => new ModInt(val);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator +(ModInt a, ModInt b)
        {
            long res = a.Data + b.Data;
            return new ModInt() { Data = res >= MOD ? res - MOD : res };
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator +(ModInt a, long b) => a.Data + b;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator -(ModInt a, ModInt b)
        {
            long res = a.Data - b.Data;
            return new ModInt() { Data = res < 0 ? res + MOD : res };
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator -(ModInt a, long b) => a.Data - b;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator *(ModInt a, ModInt b) => new ModInt() { Data = a.Data * b.Data % MOD };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static ModInt operator /(ModInt a, ModInt b) => a.Data * GetInverse(b);

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override string ToString() => Data.ToString();

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static long GetInverse(long a)
        {
            long div;
            long p = MOD;
            long x1 = 1, y1 = 0, x2 = 0, y2 = 1;
            while (true)
            {
                if (p == 1) return x2;
                div = a / p;
                x1 -= x2 * div;
                y1 -= y2 * div;
                a %= p;
                if (a == 1) return x1;
                div = p / a;
                x2 -= x1 * div;
                y2 -= y1 * div;
                p %= a;
            }
        }
    }

}