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

public class Main {

	static MyReader sc = new MyReader();
	static PrintWriter out = new PrintWriter(System.out);

	public static void main(String[] args) throws IOException {
		solve();
		out.close();
	}

	static double calc(List<Pair> list,List<Pair> newlist){
		if(list.isEmpty()){
			int x = newlist.get(0).getX();
			int y = newlist.get(0).getY();
			double ret = 0;
			for(int i=1;i<newlist.size();i++){
				int nx = newlist.get(i).getX();
				int ny = newlist.get(i).getY();
				ret += Math.sqrt(Math.pow(nx-x,2)+Math.pow(ny-y,2));
			}
			return ret;
		}else{
			double ret = 0;
			for(int i=0;i<list.size();i++){
				List<Pair> tmp1 = new ArrayList<>(list);
				List<Pair> tmp2 = new ArrayList<>(newlist);
				tmp2.add(tmp1.get(i));
				tmp1.remove(i);
				ret += calc(tmp1,tmp2);
			}
			return ret;
		}
	}


	static void solve() throws IOException {
		int n = sc.nextInt();
		List<Pair> list = new ArrayList<>();
		for(int i=0;i<n;i++){
			list.add(new Pair(sc.nextInt(),sc.nextInt()));
		}
		int div = 1;
		for(int i=1;i<n+1;i++){
			div *= i;
		}
		out.println((calc(list,new ArrayList<>())/div));
	}


	//GCD
	static int GCD(int n, int m) {
		if(m > n)
			return GCD(m, n);
		if(n % m == 0)
			return m;
		else {
			return GCD(m, n % m);
		}
	}

}

class Pair{
	private int x;
	private int y;
	protected Pair(int i,int j){
		x = i;
		y = j;
	}
	public int getX(){
		return x;
	}
	public int getY(){
		return y;
	}
}

class MyReader {
	private static BufferedReader in = new BufferedReader(new InputStreamReader(System.in));
	private static String[] line = new String[0];
	private static int num = 0;

	protected MyReader() {

	}

	private void nextline() throws IOException {
		String s = in.readLine();
		line = s.split(" ");
	}

	protected String next() throws IOException {
		if(num == line.length) {
			nextline();
			num = 0;
		}
		String ret = line[num];
		num++;
		return ret;
	}

	protected int nextInt() throws IOException {
		if(num == line.length) {
			nextline();
			num = 0;
		}
		int ret = Integer.valueOf(line[num]);
		num++;
		return ret;
	}

	protected long nextLong() throws IOException {
		if(num == line.length) {
			nextline();
			num = 0;
		}
		long ret = Long.valueOf(line[num]);
		num++;
		return ret;
	}

	protected double nextDouble() throws IOException {
		if(num == line.length) {
			nextline();
			num = 0;
		}
		double ret = Double.valueOf(line[num]);
		num++;
		return ret;
	}

	protected char nextChar() throws IOException {
		if(num == line.length) {
			nextline();
			num = 0;
		}
		char ret = line[num].toCharArray()[0];
		num++;
		return ret;
	}
}

// Combination
class Combination {
	public static int MAX = 510000;
	public static int MOD = 1000000007;

	public long[] f = new long[MAX];
	public long[] finv = new long[MAX];
	public long[] inv = new long[MAX];

	protected Combination() {
		f[0] = f[1] = 1;
		finv[0] = finv[1] = 1;
		inv[1] = 1;
		for(int i = 2; i < MAX; i++) {
			f[i] = f[i - 1] * i % MOD;
			inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
			finv[i] = finv[i - 1] * inv[i] % MOD;
		}
	}

	public long com(int n, int k) {
		if(n < k)
			return 0;
		if(n < 0 || k < 0)
			return 0;
		return f[n] * (finv[k] * finv[n - k] % MOD) % MOD;
	}

	public long inverse(int n) {
		return inv[n];
	}

	public long factorial(int n) {
		return f[n];
	}
}

//Tree
class Tree{
	private int n;
	private int root;
	private List<Integer>[] v;
	private int[] dist;
	private int[] size;

	protected Tree(int n){
		this.n = n;
		v = new ArrayList[n];
		for(int i=0;i<n;i++){
			v[i] = new ArrayList<>();
		}
		dist = new int[n];
		size = new int[n];
	}

	public void add(int i, int j){
		v[i].add(j);
		v[j].add(i);
	}

	public List<Integer> getV(int num){
		return v[num];
	}
	public void setV(List<Integer> tmp, int num){
		v[num] = new ArrayList<>(tmp);
	}
	public void setRoot(int num){
		root = num;
	}
	public int getRoot(){
		return root;
	}
	public int getDist(int i){
		return dist[i];
	}
	public int getSize(int i){
		return size[i];
	}

	public void distCalc(){
		size[root] = calc(root, -1, 0);
	}
	public void distCalc(int i){
		root = i;
		size[root] = calc(root, -1, 0);
	}
	private int calc(int here, int parent, int num){
		dist[here] = num;
		int s = 1;
		for(Integer e : v[here]){
			if(e!=parent) s += calc(e, here, num+1);
		}
		size[here] = s;
		return s;
	}
}


// Graph
class Graph {
	private int n;
	private List<Integer>[] v;
	private int[][] dist;

	protected Graph(int num) {
		n = num;
		v = new ArrayList[n];
		for(int i = 0; i < n; i++) {
			v[i] = new ArrayList<>();
		}
		dist = new int[n][n];
		for(int i = 0; i < n; i++) {
			for(int j = 0; j < n; j++) {
				if(i == j)
					dist[i][j] = 0;
				else
					dist[i][j] = 10000000;
			}
		}
	}

	public void add(int i, int j) {
		v[i].add(j);
		v[j].add(i);
		dist[i][j] = 1;
		dist[j][i] = 1;
	}

	public void add(int i, int j, int c){
		v[i].add(j);
		v[j].add(i);
		dist[i][j] = c;
		dist[j][i] = c;
	}

	public List<Integer> getV(int i){
		return v[i];
	}

	public int bfs(int i, int j) {
		boolean[] check = new boolean[n];
		Arrays.fill(check, false);
		check[i] = true;
		Queue<Integer> queue = new ArrayDeque<>();
		queue.add(i);
		while(!queue.isEmpty()) {
			int tmp = queue.poll();
			for(Integer e : v[tmp]) {
				if(!check[e]) {
					dist[i][e] = dist[i][tmp] + 1;
					queue.add(e);
				}
			}
		}
		return dist[i][j];
	}

	public void warshalFloyd() {
		for(int k = 0; k < n; k++) {
			for(int i = 0; i < n; i++) {
				for(int j = 0; j < n; j++) {
					dist[i][j] = Math.min(dist[i][j], dist[i][k] + dist[k][j]);
				}
			}
		}
	}

	public int getDist(int i, int j){
		return dist[i][j];
	}

}

// union-find木
class UnionFind {
	private int[] parent;
	private int[] depth;
	private int[] size;

	protected UnionFind(int n) {
		super();
		parent = new int[n];
		for(int i = 0; i < n; i++) {
			parent[i] = i;
		}
		depth = new int[n];
		size = new int[n];
		for(int i = 0; i < n; i++) {
			depth[i] = 1;
			size[i] = 1;
		}
	}

	public int parent(int n) {
		if(parent[n] == n) {
			return n;
		} else {
			return parent(parent[n]);
		}
	}

	public void add(int a, int b) {
		int pa = parent(a);
		int pb = parent(b);
		if(pa != pb) {
			if(depth[pa] > depth[pb]) {
				parent[pb] = pa;
				size[pa] += pb;
			} else if(depth[pa] == depth[pb]) {
				parent[pb] = pa;
				size[pa] += pb;
				depth[pa]++;
			} else {
				parent[pa] = pb;
				size[pb] += pa;
			}
		}
	}

	public int size(int n) {
		return size[parent(n)];
	}

}