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.IO;
using System.Linq;
using static System.Math;
using System.Text;
using System.Threading;
using System.Globalization;
using System.Runtime.CompilerServices;
using Library;

namespace Program
{
    public static class ABC145C
    {
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static public void Solve()
        {
            var N = NN;
            var xy = Repeat(0, N).Select(_ => new { x = NN, y = NN }).ToArray();
            double ans = 0.0;
            var allkeiro = LIB_Math.MakePermutation(N).ToList();
            foreach (var item in allkeiro)
            {
                var thisThist = 0.0;
                for (var i = 1; i < item.Count; i++)
                {
                    var dist = Sqrt((xy[item[i - 1]].x - xy[item[i]].x) * (xy[item[i - 1]].x - xy[item[i]].x) + (xy[item[i - 1]].y - xy[item[i]].y) * (xy[item[i - 1]].y - xy[item[i]].y));
                    thisThist += dist;
                }
                ans += thisThist / allkeiro.Count;
            }
            Console.WriteLine(ans);
        }
        static class Console_
        {
            static Queue<string> param = new Queue<string>();
            public static string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static public void Main(string[] args) { if (args.Length == 0) { Console.SetOut(new Printer(Console.OpenStandardOutput())); } var t = new Thread(Solve, 134217728); t.Start(); t.Join(); Console.Out.Flush(); }
        static long NN => long.Parse(Console_.NextString());
        static double ND => double.Parse(Console_.NextString());
        static string NS => Console_.NextString();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static IEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => x.OrderBy(_ => xorshift);
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
        static Func<TR> Lambda<TR>(Func<Func<TR>, TR> f) { Func<TR> t = () => default(TR); return t = () => f(t); }
        static Func<T1, TR> Lambda<T1, TR>(Func<T1, Func<T1, TR>, TR> f) { Func<T1, TR> t = x1 => default(TR); return t = x1 => f(x1, t); }
        static Func<T1, T2, TR> Lambda<T1, T2, TR>(Func<T1, T2, Func<T1, T2, TR>, TR> f) { Func<T1, T2, TR> t = (x1, x2) => default(TR); return t = (x1, x2) => f(x1, x2, t); }
        static Func<T1, T2, T3, TR> Lambda<T1, T2, T3, TR>(Func<T1, T2, T3, Func<T1, T2, T3, TR>, TR> f) { Func<T1, T2, T3, TR> t = (x1, x2, x3) => default(TR); return t = (x1, x2, x3) => f(x1, x2, x3, t); }
        static Func<T1, T2, T3, T4, TR> Lambda<T1, T2, T3, T4, TR>(Func<T1, T2, T3, T4, Func<T1, T2, T3, T4, TR>, TR> f) { Func<T1, T2, T3, T4, TR> t = (x1, x2, x3, x4) => default(TR); return t = (x1, x2, x3, x4) => f(x1, x2, x3, x4, t); }
    }
}
namespace Library {
    class LIB_Math
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Primes(long x)
        {
            if (x < 2) yield break;
            yield return 2;
            var halfx = x / 2;
            var table = new bool[halfx + 1];
            var max = (long)(Math.Sqrt(x) / 2);
            for (long i = 1; i <= max; ++i)
            {
                if (table[i]) continue;
                var add = 2 * i + 1;
                yield return add;
                for (long j = 2 * i * (i + 1); j <= halfx; j += add)
                    table[j] = true;
            }
            for (long i = max + 1; i <= halfx; ++i)
                if (!table[i] && 2 * i + 1 <= x)
                    yield return 2 * i + 1;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Factors(long x)
        {
            if (x < 2) yield break;
            while (x % 2 == 0)
            {
                x /= 2; yield return 2;
            }
            var max = (long)Math.Sqrt(x);
            for (long i = 3; i <= max; i += 2)
            {
                while (x % i == 0)
                {
                    x /= i; yield return i;
                }
            }
            if (x != 1) yield return x;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Divisor(long x)
        {
            if (x < 1) yield break;
            var max = (long)Math.Sqrt(x);
            for (long i = 1; i <= max; ++i)
            {
                if (x % i != 0) continue;
                yield return i;
                if (i != x / i) yield return x / i;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long GCD(long a, long b)
        {
            while (b > 0)
            {
                var tmp = b;
                b = a % b;
                a = tmp;
            }
            return a;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long LCM(long a, long b) => a * b / GCD(a, b);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Pow(long x, long y)
        {
            long a = 1;
            while (y != 0)
            {
                if ((y & 1) == 1) a *= x;
                x *= x;
                y >>= 1;
            }
            return a;
        }
        static List<long> _fact = new List<long>() { 1 };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void Build(long n)
        {
            if (n >= _fact.Count)
                for (int i = _fact.Count; i <= n; ++i) _fact.Add(_fact[i - 1] * i);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Comb(long n, long k)
        {
            Build(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)] / _fact[(int)k];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Perm(long n, long k)
        {
            Build(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<List<int>> MakePermutation(long n, bool zeroIndexed = true)
        {
            if (n <= 0) throw new Exception();
            var c = new int[n];
            var a = new int[n];
            if (!zeroIndexed) a[0] = 1;
            for (var i = 1; i < n; i++) a[i] = a[i - 1] + 1;
            yield return new List<int>(a);
            for (var i = 0; i < n;)
            {
                if (c[i] < i)
                {
                    if (i % 2 == 0)
                    {
                        var t = a[0]; a[0] = a[i]; a[i] = t;
                    }
                    else
                    {
                        var t = a[c[i]]; a[c[i]] = a[i]; a[i] = t;
                    }
                    yield return new List<int>(a);
                    ++c[i];
                    i = 0;
                }
                else
                {
                    c[i] = 0;
                    ++i;
                }
            }
        }
    }
}