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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.*;
import java.io.*;

public class Main {
	private static void solve(){
	//Implement solution here.
		int n = ni();
		double[][] xy = new double[n][2];
		for(int i = 0; i < n; i++) {
			xy[i][0] = nd();
			xy[i][1] = nd();
		}
		double lengsum = 0;
		
		for(int i = 0; i < n; i++) {
			for(int j = i + 1; j < n; j++) {
				double ax = xy[i][0];
				double ay = xy[i][1];
				double bx = xy[j][0];
				double by = xy[j][1];
 				lengsum += Math.pow(Math.pow(ax - bx, 2) + Math.pow(ay - by, 2), 0.5);
 				//System.out.println(lengsum);
			}
		}
		
		int all = 1;
		int section = n - 1;
		int choice = n * (n - 1) / 2;
		int tmp = n;
		while(tmp > 1) {
			all = all * tmp;
			tmp--;
		}
		int b = all * section / choice;
		System.out.println(lengsum * b / all);

	}

    //Switch input source (stdin/file)
	private static BufferedReader br  = new BufferedReader(new InputStreamReader(System.in));
	public static void main(String[] args){
        String debugDataPath = System.getenv("DD_PATH");        
        if(debugDataPath != null){
            try{
            	br = new BufferedReader(new InputStreamReader(new FileInputStream(new File(debugDataPath))));
            }catch(Exception e){
                throw new RuntimeException(e);
            }
        }
        solve();
	}

    //Input read utility
	private static StringTokenizer tokenizer = null;
	public static String next() {
		while (tokenizer == null || !tokenizer.hasMoreTokens()) {
			try {
				tokenizer = new StringTokenizer(br.readLine());
			} catch (Exception e) {
				throw new RuntimeException(e);
			}
		}
		return tokenizer.nextToken();
	}
    //Get next single int
	private static int ni() {
		return Integer.parseInt(next());
	}
    //Get next single long
	private static long nl() {
		return Long.parseLong(next());
	}
    //Get next single double
	private static double nd() {
		return Double.parseDouble(next());
	}
    //Get next int array from one line
	private static int[] na(int n) {
		int[] a = new int[n];
		for (int i = 0; i < n; i++)
			a[i] = ni();
		return a;
	}
	//Get next char array from one line
	private static char[] ns() {
		return next().toCharArray();
	}
    //Get next long array from one line
	private static long[] nal(int n) {
		long[] a = new long[n];
		for (int i = 0; i < n; i++)
			a[i] = nl();
		return a;
	}
}