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: Accepted)

import java.util.*;
import java.io.*;
import java.math.BigInteger;

 
class Main implements Runnable {
  // static Scanner in;
  static FastReader in;
  static PrintWriter out;
  static int[][] dirs8 = {{1, 0}, {-1, 0}, {0, 1}, {0, -1}, {1, 1}, {-1, -1}, {-1, 1}, {1, -1}};
  static int[][] dirs = {{0, -1}, {1, 0}, {0, +1}, {-1, 0}};
 
  static int[][] E;
  // static List<int[]>[] adj;
  static List<Integer>[] adj;
  static int[] vis;
 
  static long INFL = (long) 1e15 + 7;
  static int INF = (int) 1e9 + 7;
  static int mod = 998244353;
  
  static int T, n, m, k, count, cur, w;
  static boolean flag;
  // static int[] a, b;
  static int[][] map;

  static int[] par;
  static double[][][] p;
  static double[][][] ev;

  static void solve() throws Exception {
    n = in.nextInt();
    p = new double[n + 1][n + 1][n + 1];
    ev = new double[n + 1][n + 1][n + 1];

    // ev[i][j][k]: E{# of ops to get i 1-dish, j 2-dish, k 3-dish}
    // p[i][j][k]: P{probability to get to i 1-dish, j 2-dish, k 3-dish}

    int[] start = new int[4];
    for (int i = 0; i < n; i++) {
      start[in.nextInt()]++;
    }

    p[start[1]][start[2]][start[3]] = 1;
    
    for (int c = n; c >= 0; c--) {
      for (int b = n; b >= 0; b--) {
        for (int a = n; a >= 0; a--) {
          int left = a + b + c;
          if (left == 0 || left > n) {continue;}
          double pWaste = (double) (n - left) / n;
          double evWaste = pWaste / (1 - pWaste);
          ev[a][b][c] += (evWaste + 1) * p[a][b][c];

          if (a != 0) {
            double p_go = ((double) a) / (a + b + c);
            p[a-1][b][c] += p[a][b][c] * p_go;
            ev[a-1][b][c] += ev[a][b][c] * p_go;
          }

          if (b != 0) {
            double p_go = ((double) b) / (a + b + c);
            p[a+1][b - 1][c] += p[a][b][c] * p_go;
            ev[a+1][b - 1][c] += ev[a][b][c] * p_go;
          }

          if (c != 0) {
            double p_go = ((double) c) / (a + b + c);
            p[a][b + 1][c - 1] += p[a][b][c] * p_go;
            ev[a][b + 1][c - 1] += ev[a][b][c] * p_go;
          }
        }
      }
    }

    out.println(ev[0][0][0]);
  }


  public static void main(String[] args) throws Exception {
 
    // in = new Scanner(new BufferedReader(new InputStreamReader(System.in)));
    in = new FastReader();
    out = new PrintWriter(System.out);
 
    // int numOfTests = in.nextInt();
    // for (int caseNum = 1; caseNum <= numOfTests; caseNum++) {
    //   // Thread t = new Thread(null, new Main(), "Main", 1 << 28);
    //   // t.start();
    //   // t.join();
    //   solve();
    // }
 
    // // Thread t = new Thread(null, new Main(), "Main", 1 << 28);
    // // t.start();
    // // t.join();
    solve();
 
    out.flush();
    out.close();
  }
 
  @Override
  public void run() {
    try {
      solve();
    } catch (Exception e) {
      e.printStackTrace(); 
    }
  }
 
  static class FastReader {
    public BufferedReader br; 
    StringTokenizer st; 
    public FastReader() {br = new BufferedReader(new InputStreamReader(System.in));} 
    String next() { 
      while (st == null || !st.hasMoreElements()) { 
          try {st = new StringTokenizer(br.readLine());} 
          catch (IOException e) {e.printStackTrace();} 
      } 
      return st.nextToken(); 
    } 
    int nextInt() {return Integer.parseInt(next());} 
    long nextLong() {return Long.parseLong(next());} 
    double nextDouble() {return Double.parseDouble(next());} 
    String nextLine() {
      String str = ""; 
        try {str = br.readLine();} 
        catch (IOException e) {e.printStackTrace();} 
        return str; 
    }
  }

  static void as(boolean result) throws Exception {
    if (!result) {
      throw new Exception();
    }
  }

  static int modInverse(int a, int m) { 
    int m0 = m; 
    int y = 0, x = 1; 
    if (m == 1) {return 0;} 
    while (a > 1) { 
      // q is quotient 
      int q = a / m; 
      int t = m; 
      // m is remainder now, process 
      // same as Euclid's algo 
      m = a % m; 
      a = t; 
      t = y; 
      // Update x and y 
      y = x - q * y; 
      x = t; 
    } 
    // Make x positive 
    if (x < 0) 
      x += m0; 
    return x; 
  } 
}