                    ABCD
                    CABD
                    ACDB
                    DACB
                    BCDA
                    ACBD
                    ADCB
                    CDAB
                    DABC
                    BCAD
                    CADB
                    CDBA
                    CBAD
                    ABDC
                    ADBC
                    BDCA
                    DCBA
                    BACD
                    BADC
                    BDAC
                    CBDA
                    DBCA
                    DCAB

Listed above are   all-but-one   of the permutations of the symbols   A,   B,   C,   and   D,   except   for one permutation that's   not   listed.



Task

Find that missing permutation.



Methods

 Obvious method: 
        enumerate all permutations of   A,  B,  C,  and  D,  
        and then look for the missing permutation. 

 alternate method:
        Hint:  if all permutations were shown above,  how many 
        times would  A  appear in each position?     
        What is the  parity  of this number?

 another alternate method:
        Hint:  if you add up the letter values of each column, 
        does a missing letter   A,  B,  C,  and  D   from each
        column cause the total value for each column to be unique?



Related task

   Permutations)


using System;
using System.Collections.Generic;
 
namespace MissingPermutation
{
    class Program
    {
        static void Main()
        {
            string[] given = new string[] { "ABCD", "CABD", "ACDB", "DACB", 
                                            "BCDA", "ACBD", "ADCB", "CDAB", 
                                            "DABC", "BCAD", "CADB", "CDBA", 
                                            "CBAD", "ABDC", "ADBC", "BDCA", 
                                            "DCBA", "BACD", "BADC", "BDAC", 
                                            "CBDA", "DBCA", "DCAB" };
 
            List<string> result = new List<string>();
            permuteString(ref result, "", "ABCD");
 
            foreach (string a in result)            
                if (Array.IndexOf(given, a) == -1)
                    Console.WriteLine(a + " is a missing Permutation");
        }
 
        public static void permuteString(ref List<string> result, string beginningString, string endingString)
        {
            if (endingString.Length <= 1)
            {                 
                result.Add(beginningString + endingString);
            }
            else
            {
                for (int i = 0; i < endingString.Length; i++)
                {                     
                    string newString = endingString.Substring(0, i) + endingString.Substring(i + 1);
                    permuteString(ref result, beginningString + (endingString.ToCharArray())[i], newString);                    
                }
            }
        }
    }
}