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 EducationalDPContest.Questions;
using EducationalDPContest.Extensions;
using EducationalDPContest.Algorithms;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;

namespace EducationalDPContest.Questions
{
    public class QuestionJ : AtCoderQuestionBase
    {
        public override IEnumerable<object> Solve(TextReader inputStream)
        {
            var n = inputStream.ReadInt();
            var sushis = inputStream.ReadIntArray();

            yield return GetExpectedValue(sushis.Length, sushis.Count(i => i == 1), sushis.Count(i => i == 2), sushis.Count(i => i == 3));
        }

        double[,,] dpCache = new double[310, 310, 310];

        double GetExpectedValue(int all, int one, int two, int three)
        {
            if (one == 0 && two == 0 && three == 0)
            {
                return 0;
            }
            if (dpCache[one, two, three] > 0)
            {
                return dpCache[one, two, three];
            }

            double expected = all;
            if (one > 0)
            {
                expected += GetExpectedValue(all, one - 1, two, three) * one;
            }
            if (two > 0)
            {
                expected += GetExpectedValue(all, one + 1, two - 1, three) * two;
            }
            if (three > 0)
            {
                expected += GetExpectedValue(all, one, two + 1, three - 1) * three;
            }
            expected /= one + two + three;

            return dpCache[one, two, three] = expected;
        }
    }
}


namespace EducationalDPContest
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionJ();    // 問題に合わせて書き換え
            var answers = question.Solve(Console.In);
            foreach (var answer in answers)
            {
                Console.WriteLine(answer);
            }
        }
    }
}

#region Base Classes

namespace EducationalDPContest.Questions
{

    public interface IAtCoderQuestion
    {
        IEnumerable<object> Solve(string input);
        IEnumerable<object> Solve(TextReader inputStream);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public IEnumerable<object> Solve(string input)
        {
            var stream = new MemoryStream(Encoding.Unicode.GetBytes(input));
            var reader = new StreamReader(stream, Encoding.Unicode);

            return Solve(reader);
        }

        public abstract IEnumerable<object> Solve(TextReader inputStream);

        protected void UpdateWhenSmall<T>(ref T value, T other) where T : IComparable<T>
        {
            if (other.CompareTo(value) < 0)
            {
                value = other;
            }
        }

        protected void UpdateWhenLarge<T>(ref T value, T other) where T : IComparable<T>
        {
            if (other.CompareTo(value) > 0)
            {
                value = other;
            }
        }
    }

}

#endregion

#region Algorithms

namespace EducationalDPContest.Algorithms
{
    public struct Modular : IEquatable<Modular>, IComparable<Modular>
    {
        private const int _defaultMod = 1000000007;
        public int Value { get; }
        public int Mod { get; }     // 配列とかで引数なしコンストラクタが呼ばれると0で初期化されて死ぬ（ガイドライン違反）（int?にすると割り算が2倍遅くなる）（諦め）（運用でカバー）

        public Modular(long value, int mod = _defaultMod)
        {
            if (mod < 2 || mod > 1073741789)
            {
                // 1073741789はint.MaxValue / 2 = 1073741823以下の最大の素数
                throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上1073741789以下の素数でなければなりません。");
            }
            Mod = mod;

            if (value >= 0 && value < mod)
            {
                Value = (int)value;
            }
            else
            {
                Value = (int)(value % mod);
                if (Value < 0)
                {
                    Value += mod;
                }
            }
        }

        public static Modular operator +(Modular a, Modular b)
        {
            CheckModEquals(a, b);

            var result = a.Value + b.Value;
            if (result > a.Mod)
            {
                result -= a.Mod;    // 剰余演算を避ける
            }
            return new Modular(result, a.Mod);
        }

        public static Modular operator -(Modular a, Modular b)
        {
            CheckModEquals(a, b);

            var result = a.Value - b.Value;
            if (result < 0)
            {
                result += a.Mod;    // 剰余演算を避ける
            }
            return new Modular(result, a.Mod);
        }

        public static Modular operator *(Modular a, Modular b)
        {
            CheckModEquals(a, b);
            return new Modular((long)a.Value * b.Value, a.Mod);
        }

        public static Modular operator /(Modular a, Modular b)
        {
            CheckModEquals(a, b);
            return a * Pow(b, a.Mod - 2);
        }

        // 需要は不明だけど一応
        public static bool operator ==(Modular left, Modular right) => left.Equals(right);
        public static bool operator !=(Modular left, Modular right) => !(left == right);
        public static bool operator <(Modular left, Modular right) => left.CompareTo(right) < 0;
        public static bool operator <=(Modular left, Modular right) => left.CompareTo(right) <= 0;
        public static bool operator >(Modular left, Modular right) => left.CompareTo(right) > 0;
        public static bool operator >=(Modular left, Modular right) => left.CompareTo(right) >= 0;

        public static explicit operator int(Modular a) => a.Value;
        public static explicit operator long(Modular a) => a.Value;

        public static Modular Pow(Modular a, int n)
        {
            if (n < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"べき指数{nameof(n)}は0以上の整数でなければなりません。");
            }
            switch (n)
            {
                case 0:
                    return new Modular(1, a.Mod);
                case 1:
                    return a;
                default:
                    var p = Pow(a, n >> 1);             // m / 2
                    return p * p * Pow(a, n & 0x01);    // m % 2
            }
        }

        private static Dictionary<int, List<int>> FactorialCache = new Dictionary<int, List<int>>();
        private static Dictionary<long, List<int>> PermutationCache = new Dictionary<long, List<int>>();

        public static Modular Factorial(int n, int mod = _defaultMod)
        {
            if (n < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません。");
            }
            if (mod < 2)
            {
                throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
            }

            if (!FactorialCache.ContainsKey(mod))
            {
                FactorialCache.Add(mod, new List<int>() { 1 });
            }

            var cache = FactorialCache[mod];
            for (int i = cache.Count; i <= n; i++)  // Countが1（0!までキャッシュ済み）のとき1!～n!まで計算
            {
                cache.Add((int)((long)cache[i - 1] * i % mod));
            }
            return new Modular(cache[n], mod);
        }

        public static Modular Permutation(int n, int r, int mod = _defaultMod)
        {
            CheckNR(n, r);

            if (!PermutationCache.ContainsKey(Concat(mod, n)))
            {
                PermutationCache.Add(Concat(mod, n), new List<int>() { 1 });
            }

            var cache = PermutationCache[Concat(mod, n)];
            for (int i = cache.Count; i <= r; i++)  // Countが1（nP0までキャッシュ済み）のときnP1～nPrまで計算
            {
                cache.Add((int)((long)cache[i - 1] * (n - (i - 1)) % mod));
            }
            return new Modular(cache[r], mod);
        }

        public static Modular Combination(int n, int r, int mod = _defaultMod)
        {
            CheckNR(n, r);
            r = Math.Min(r, n - r);
            return n == r ? new Modular(1, mod) : Permutation(n, r, mod) / Factorial(r, mod);   // nCr = n! / (n-r)!r! = nPr / n!
        }

        public static Modular[] CreateArray(int length, int mod = _defaultMod) => Enumerable.Repeat(new Modular(0, mod), length).ToArray();

        private static void CheckModEquals(Modular a, Modular b)
        {
            if (a.Mod != b.Mod)
            {
                throw new ArgumentException($"{nameof(a)}, {nameof(b)}", $"両者の法{nameof(Mod)}は等しくなければなりません。");
            }
        }

        private static void CheckNR(int n, int r)
        {
            if (n <= 0)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は正の整数でなければなりません。");
            }
            if (r < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(r), $"{nameof(r)}は0以上の整数でなければなりません。");
            }
            if (n < r)
            {
                throw new ArgumentOutOfRangeException($"{nameof(n)},{nameof(r)}", $"{nameof(r)}は{nameof(n)}以下でなければなりません。");
            }
        }

        public override string ToString() => $"{Value} (mod {Mod})";

        public override bool Equals(object obj) => obj is Modular ? Equals((Modular)obj) : false;

        public bool Equals(Modular other) => Value == other.Value && Mod == other.Mod;

        public int CompareTo(Modular other)
        {
            CheckModEquals(this, other);
            return Value.CompareTo(other.Value);
        }

        public override int GetHashCode() => Concat(Value, Mod).GetHashCode();

        static long Concat(int a, int b) => (a << 32) | b;
    }
}

#endregion

#region Collections

namespace EducationalDPContest.Collections
{
    public static class SearchExtension
    {
        public static int GetGreaterEqualIndex<T>(this T[] span, T minValue) where T : IComparable<T>
        {
            int ng = -1;
            int ok = span.Length;

            return BoundaryBinarySearch(span, v => v.CompareTo(minValue) >= 0, ng, ok);
        }

        public static int GetSmallerEqualIndex<T>(this T[] span, T maxValue) where T : IComparable<T>
        {
            int ng = span.Length;
            int ok = -1;

            return BoundaryBinarySearch(span, v => v.CompareTo(maxValue) <= 0, ng, ok);
        }

        private static int BoundaryBinarySearch<T>(T[] span, Predicate<T> predicate, int ng, int ok)
        {
            // めぐる式二分探索
            // Span.BinarySearchだとできそうでできない（lower_boundがダメ）
            while (Math.Abs(ok - ng) > 1)
            {
                int mid = (ok + ng) / 2;

                if (predicate(span[mid]))
                {
                    ok = mid;
                }
                else
                {
                    ng = mid;
                }
            }
            return ok;
        }
    }
}

#endregion

#region Extensions

namespace EducationalDPContest.Extensions
{
    internal static class TextReaderExtensions
    {
        internal static int ReadInt(this TextReader reader) => int.Parse(ReadString(reader));
        internal static long ReadLong(this TextReader reader) => long.Parse(ReadString(reader));
        internal static double ReadDouble(this TextReader reader) => double.Parse(ReadString(reader));
        internal static string ReadString(this TextReader reader) => reader.ReadLine();

        internal static int[] ReadIntArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(int.Parse).ToArray();
        internal static long[] ReadLongArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(long.Parse).ToArray();
        internal static double[] ReadDoubleArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(double.Parse).ToArray();
        internal static string[] ReadStringArray(this TextReader reader, char separator = ' ') => reader.ReadLine().Split(separator);
    }
}

#endregion