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.ArrayList;
import java.util.HashSet;
import java.util.Scanner;
class Main{


	static final int MOD = 1000000007;


	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);			//文字の入力
		int n = sc.nextInt();
		ArrayList<Town> towns = new ArrayList<Town>();
		for(int i = 0;i < n;i++){
			int x = sc.nextInt();
			int y = sc.nextInt();
			Town town = new Town(i, x, y);
			towns.add(town);
		}
		
	ArrayList<Double> ans = new ArrayList<Double>();
	for(int i = 0;i < n;i++){
		Town town = towns.get(i);
		HashSet<Integer> set = new HashSet<>();
		set.add(i);
		dfs(ans, towns, i, town.x, town.y, set, n, (double)0);
	}
	
	
	double num = 0;
	
	
	for(int i = 0;i < ans.size();i++){
		num += ans.get(i);
	}
//	System.out.println(num);
	System.out.println(num/ans.size());

		


		




	}
	
	
	static void dfs(ArrayList<Double> ans,ArrayList<Town> towns,int index,int cx,int cy,HashSet<Integer> used,int n,double score){
		if(used.size() == n){
			ans.add(score);
			return;
		}
//		System.out.println(score);
		for(int i = 0;i < n;i++){
			if(!used.contains(i)){
				used.add(i);
				Town town = towns.get(i);
				
				dfs(ans, towns,i, town.x, town.y, used, n, score+Math.sqrt(Math.abs(cx-town.x)*Math.abs(cx-town.x) + Math.abs(cy-town.y)*Math.abs(cy-town.y)));
				used.remove(i);
			}
		}
		
		
		
	}

	static void sieve(ArrayList<Long> list){
		int n = 50000000;
		boolean[] is_prime = new boolean[n+1];
		int[] prime = new int[n];
		for(int i= 0;i <=n;i++){
			is_prime[i] = true;
		}
		list.add((long) 1);
		for(int i = 2;i <= n;i++){
			if(is_prime[i]){
				list.add((long)i);
				for(int j = 2*i;j <= n;j+=i){
					is_prime[j] = false;
				}
			}
		}



	}




	static int upperbond(int k,int min,int[] data){		//kより小さい最大の値をdataの中から探し、返す二分探索
		int max = data.length;
		//		if(min == data.length-1){
		//			if(data[min] > k){
		//				return min;
		//			}
		//		}

		while(max-min > 1){
			int mid = (max+min)/2;
			if(data[mid] >= k){
				max = mid;
			}else{
				min = mid;
			}
		}

		return max;
	}

	static int lowerbond(int k,int min,int[] data){		//kより大きい最小の値をdataの中から探し、返す二分探索
		int max = data.length;

		while(max-min > 1){
			int mid = (max+min)/2;
			if(data[mid] >= k){
				max = mid;
			}else{
				min = mid;
			}
		}
		return min;
	}



	static int gcd(int a,int b){				//最大公約数を返す
		if(b == 0){
			return a;
		}else{
			return gcd(b, a%b);
		}
	}
	static long gcd(long a,long b){
		if(b == 0){
			return a;
		}else{
			return gcd(b, a%b);
		}
	}



	static long lcm (long a, long b) {
		long g = gcd(a,b);
		return a/g*b;
	}




}

class Town{
	int index;
	int x ;
	int y;
	public Town(int index,int x,int y) {
		this.index = index;
		this.x = x;
		this.y = y;
		// TODO 自動生成されたコンストラクター・スタブ
	}
	
	
}


class Edge extends ArrayList<P>{

}

class P{
	int to;
	long cost;
	P(int to,long cost){
		this.to = to;
		this.cost = cost;
	}
}


class Pair implements Comparable{
	int from;
	int end;
	@Override
	public int compareTo(Object other) {
		Pair otherpair = (Pair)other;

		return   from- otherpair.from;
	}








}