using System;

using System.Collections.Generic;

using System.Linq;

public class Program

{

    public static void Main()

    {

        const int trials = 1000000;

        Console.WriteLine($"Trials={trials} probability not all values received after n rolls versus observed");

        var rn = new Random(1000);

        const int N = 20;   // an N-dice

        var active = new List<int>()  { 0};

        for (var trial = 0; trial < trials; trial++)

        {

            var bComplete = false;

            var rolls = new int[N];

            var roll = 0;

            bool done;

            active[0]++;

            do

            {

                if (active.Count-1 < ++roll ) active.Add(0);

                rolls[rn.Next(0, N)]++;

                done = !rolls.Any(x => x == 0);

                if (!done)

                {

                    active[roll]++;

​

                }

​

            } while(!done) ;

        }

        var sum = active.Sum(x=>x);

        var results = active.Select(( x, i)=> 

        (

        roll:i,

        observed: (double) x/ trials, 

        predicted: (double) inclusionExclusion((uint) i,N)

        )).ToArray();

​

        foreach(var result in results)

        {

            Console.WriteLine($"roll {result.roll}   predicted {result.predicted}  observed probability {result.observed}");

        }

    }

     private static double inclusionExclusion(uint trial, uint ndie)

     {

         if (trial < ndie) return 1;

         //var util = new BallsInUrns();

         double fact = 1.0 / ndie;

         double prob = 0;

         for (uint i = 1; i < ndie;i++)

         {

             prob += (double) (i % 2 == 1 ? 1:-1) * C(ndie, i) * Math.Pow((ndie - i) * fact, trial);

         }

         return prob;

     }

        public static double Factorial(uint n)

    {

        return (n > 0) ? Factorial(n - 1) * n : 1;

    }

​

    public static double C(uint n, uint r)

    {

        return Factorial(n)/ Factorial(r) / Factorial(n - r) ;

    }

}