Problem:
Score : 100 points

Problem Statement

There are N dishes, numbered 1, 2, \ldots, N.
Initially, for each i (1 \leq i \leq N), Dish i has a_i (1 \leq a_i \leq 3) pieces of sushi on it.

Taro will perform the following operation repeatedly until all the pieces of sushi are eaten:

Roll a die that shows the numbers 1, 2, \ldots, N with equal probabilities, and let i be the outcome. If there are some pieces of sushi on Dish i, eat one of them; if there is none, do nothing.

Find the expected number of times the operation is performed before all the pieces of sushi are eaten.

Constraints

All values in input are integers.

1 \leq N \leq 300

1 \leq a_i \leq 3

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 \ldots a_N

Output

Print the expected number of times the operation is performed before all the pieces of sushi are eaten.
The output is considered correct when the relative difference is not greater than 10^{-9}.

Sample Input 1

3
1 1 1

Sample Output 1

5.5

The expected number of operations before the first piece of sushi is eaten, is 1.
After that, the expected number of operations before the second sushi is eaten, is 1.5.
After that, the expected number of operations before the third sushi is eaten, is 3.
Thus, the expected total number of operations is 1 + 1.5 + 3 = 5.5.

Sample Input 2

1
3

Sample Output 2

3

Outputs such as 3.00, 3.000000003 and 2.999999997 will also be accepted.

Sample Input 3

2
1 2

Sample Output 3

4.5

Sample Input 4

10
1 3 2 3 3 2 3 2 1 3

Sample Output 4

54.48064457488221

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import javax.swing.text.AbstractDocument;
import java.io.*;
import java.util.*;

import static java.lang.Math.*;

public class Main {
    static public class pair {
        int u;
        int idx;
        int val;

        public pair(int x, int i) {
            u = x;
            idx = i;
            val = -1;
        }
    }


    static ArrayList<Integer>[] adj;
    static int[] value, weight, Arr;
    static long inf = (long) 1e16;
    static int n, m;

    static boolean[][] matrix;
    static double[][] memo;
    static long mod = (long) (1e9 + 7);

    public static boolean sub(String a, String b) {
        for (int i = 0; i < a.length(); i++) {
            if (a.charAt(i) == b.charAt(i) || a.charAt(i) == '?' || b.charAt(i) == '?')
                continue;
            return false;
        }
        return true;
    }

    public static boolean cont(String a, String b) {
        int n = b.length();
        for (int i = 0; i < a.length() - n; i++) {
            if (sub(a.substring(i, i + n), b))
                return true;
        }
        return false;
    }

    public static int inter(String a, String b) {
        int n = min(a.length(), b.length());
        if (cont(a, b) || cont(b, a))
            return n;
        while (n > 0) {
            if (sub(a.substring(a.length() - n), (b.substring(0, n))))
                return n;
            n--;
        }
        return 0;
    }

    static TreeSet<String> ts;

    public static void backtrack(int idx, int upperBound, String str) {
        if (idx == n) {
            ts.add(str);
            return;
        }
        for (int i = 0; i <= upperBound + 1; i++) {
            backtrack(idx + 1, Math.max(upperBound, i), str + (char) ('a' + i));
        }
    }

    static int w, maxvalue = 2;
    static double[] prob;
    static double mem[][][];
    static int[][][] memm;

    public static double dp(int one, int two, int three) {
        if (one == 0 && two == 0 && three == 0) return 0;
        if (mem[one][two][three] > -0.5) return mem[one][two][three];
        double ans = 0;
        double d = (n * 1d) / (one + two + three);
        int c = 0;
        if (three > 0) {
            ans += (d + dp(one, two + 1, three - 1));
            c++;
        }
        if (two > 0) {
            ans += (d + dp(one + 1, two - 1, three));
            c++;
        }
        if (one > 0) {
            ans += (d + dp(one - 1, two, three));
            c++;
        }
        return mem[one][two][three] = ans / c;
    }

    public static int count(int one, int two, int three) {
        if (one == 0 && two == 0 && three == 0) return 1;
        if (memm[one][two][three] != -1) return memm[one][two][three];
        int ans = 0;
        if (three > 0)
            ans += count(one, two + 1, three - 1);
        if (two > 0)
            ans += count(one + 1, two - 1, three);
        if (one > 0)
            ans += count(one - 1, two, three);
        return memm[one][two][three] = ans;
    }

    static PrintWriter pw;


    static int[] dp;


    public static void main(String[] args) throws Exception {

        Scanner sc = new Scanner(System.in);
        pw = new PrintWriter(System.out);
        n = sc.nextInt();
        int[] count = new int[4];
        for (int i = 0; i < n; i++) {
            int a = sc.nextInt();
            count[a]++;
        }
        memm = new int[302][302][302];
        mem = new double[302][302][302];
        for (int[][] y : memm) for (int[] x : y) Arrays.fill(x, -1);
        for (double[][] y : mem) for (double[] x : y) Arrays.fill(x, -1);
        double nem = dp(count[1], count[2], count[3]);
        int den = count(count[1], count[2], count[3]);
        System.out.println(nem + " " + den);
        pw.println(nem);
        pw.close();
    }

    public static class Scanner {
        StringTokenizer st;
        BufferedReader br;

        public Scanner(InputStream s) {
            br = new BufferedReader(new InputStreamReader(s));
        }

        public Scanner(String s) throws FileNotFoundException {
            br = new BufferedReader(new FileReader(s));

        }

        public String next() throws IOException {
            while (st == null || !st.hasMoreTokens())
                st = new StringTokenizer(br.readLine());
            return st.nextToken();
        }

        public int nextInt() throws IOException {
            return Integer.parseInt(next());
        }

        public long nextLong() throws IOException {
            return Long.parseLong(next());
        }

        public String nextLine() throws IOException {
            return br.readLine();
        }

        public double nextDouble() throws IOException {
            String x = next();
            StringBuilder sb = new StringBuilder("0");
            double res = 0, f = 1;
            boolean dec = false, neg = false;
            int start = 0;
            if (x.charAt(0) == '-') {
                neg = true;
                start++;
            }
            for (int i = start; i < x.length(); i++)
                if (x.charAt(i) == '.') {
                    res = Long.parseLong(sb.toString());
                    sb = new StringBuilder("0");
                    dec = true;
                } else {
                    sb.append(x.charAt(i));
                    if (dec)
                        f *= 10;
                }
            res += Long.parseLong(sb.toString()) / f;
            return res * (neg ? -1 : 1);
        }

        public boolean ready() throws IOException, IOException {
            return br.ready();
        }

    }
}