                    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)


# our deficient list
L :=
[ "ABCD", "CABD", "ACDB", "DACB", "BCDA",
  "ACBD", "ADCB", "CDAB", "DABC", "BCAD",
  "CADB", "CDBA", "CBAD", "ABDC", "ADBC",
  "BDCA", "DCBA", "BACD", "BADC", "BDAC",
  "CBDA", "DBCA", "DCAB" ];
 
# convert L to permutations on 1..4
u := List(L, s -> List([1..4], i -> Position("ABCD", s[i])));
 
# set difference (with all permutations)
v := Difference(PermutationsList([1..4]), u);
 
# convert back to letters
s := "ABCD";
List(v, p -> List(p, i -> s[i]));