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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Linq;
using System.Text;

namespace EDP_J
{
    class Program
    {
        static double solvedp(double[,,] dp, int n, int i, int j, int k)
        {
            if (dp[i, j, k] >= 0) return dp[i, j, k];
            if ((i + j + k) == 0) return 0;

            var ans = 0.0;
            if (i > 0) ans += solvedp(dp, n, i - 1, j, k) * i;
            if (j > 0) ans += solvedp(dp, n, i + 1, j - 1, k) * j;
            if (k > 0) ans += solvedp(dp, n, i, j + 1, k - 1) * k;

            ans += n;
            ans *= 1.0 / (i + j + k);

            dp[i, j, k] = ans;

            return ans;
        }

        static double[,,] solvedp(List<int> nums)
        {
            var n = nums.Count();

            // 残数が同じ皿は、同一視してよく、上限が3貫と決まっているので
            // 3変数に集約できる
            // [i, j, k]: 残り1貫の皿がi枚～残り3貫の皿がk枚の状態から
            // お寿司をすべてなくすために必要な操作回数の期待値
            var dp = new double[n + 1, n + 1, n + 1];
            for (int i = 0; i < dp.GetLength(0); i++)
            {
                for (int j = 0; j < dp.GetLength(1); j++)
                {
                    for (int k = 0; k < dp.GetLength(2); k++)
                    {
                        dp[i, j, k] = -1;
                    }
                }
            }

            solvedp(dp, n,
                nums.Count(x => x == 1),
                nums.Count(x => x == 2),
                nums.Count(x => x == 3));

            return dp;
        }

        static void Solve()
        {
            var n = Input.NextInt();
            var a = Input.NextInt(n);

            var dp = solvedp(a);

            var ans = dp[a.Count(x => x == 1),
                a.Count(x => x == 2),
                a.Count(x => x == 3)];

            Console.WriteLine($"{ans:F10}");
        }

        #region Competitive Template
        public static void Main(string[] args)
        {
            var needsFlushOutput = true;
            if (needsFlushOutput)
            {
                // 細かく出力しないようにする
                var sw = new System.IO.StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
                Console.SetOut(sw);
            }

            // 仮想的に標準入力をセットする
            // NextLine系を使っていると使えない
            //Input.SetText("");

            Solve();

            Console.Out.Flush();
        }

        static class Input
        {
            static char[] separator = { ' ' };
            public static bool IsEof { get; set; }
            static Queue<string> q { get; set; }

            static Input()
            {
                IsEof = false;
                q = new Queue<string>();
            }

            /// <summary>
            /// 入力予約
            /// </summary>
            /// <param name="items"></param>
            public static void SetText(IEnumerable<string> items)
            {
                foreach (var item in items)
                {
                    SetText(item);
                }
            }

            /// <summary>
            /// 入力予約
            /// </summary>
            /// <param name="s"></param>
            /// <returns></returns>
            public static bool SetText(string s)
            {
                if (s == null) return false;

                foreach (var elem in s.Trim().Split(separator, StringSplitOptions.RemoveEmptyEntries))
                {
                    q.Enqueue(elem);
                }

                return true;
            }

            /// <summary>
            /// 内部queueに入力からの値をsplitして格納する
            /// </summary>
            /// <returns></returns>
            static bool read()
            {
                var s = Console.ReadLine();
                if (s == null) return false;

                foreach (var elem in s.Trim().Split(separator, StringSplitOptions.RemoveEmptyEntries))
                {
                    q.Enqueue(elem);
                }

                if (!q.Any()) return read();

                return true;
            }

            /// <summary>
            /// 次のstringを一つ読み込む
            /// </summary>
            /// <returns></returns>
            public static string Next()
            {
                if (!q.Any())
                {
                    if (!read())
                    {
                        IsEof = true;
                        return "";
                    }
                }

                return q.Dequeue();
            }

            public static int NextInt() => int.Parse(Next());
            public static long NextLong() => long.Parse(Next());
            public static double NextDouble() => double.Parse(Next());

            public static List<string> Next(int n) => Enumerable.Range(0, n).Select(_ => Next()).ToList();
            public static List<int> NextInt(int n) => Next(n).Select(x => int.Parse(x)).ToList();
            public static List<long> NextLong(int n) => Next(n).Select(x => long.Parse(x)).ToList();
            public static List<double> NextDouble(int n) => Next(n).Select(x => double.Parse(x)).ToList();

            public static List<string> NextLine() => Console.ReadLine().Trim().Split(separator, StringSplitOptions.RemoveEmptyEntries).ToList();
            public static List<int> NextIntLine() => NextLine().Select(x => int.Parse(x)).ToList();
            public static List<long> NextLongLine() => NextLine().Select(x => long.Parse(x)).ToList();
            public static List<double> NextDoubleLine() => NextLine().Select(x => double.Parse(x)).ToList();
        }
        #endregion
    }
}