Question

Suppose that the name of a file in a computer directory consists of three digits followed by two lowercase letters and each digit is 0,1 , or 2 , and each letter is either a or b. List the name of these files in lexicographic order, where we order letters using the usual alphabetic order of letters.

Short answer

The list in lexicographic order is:

\(\begin{array}{l}{\rm{000 a a, 000 a b, 000 b a, 000 b b, 001 a a, 001 a b, 001 b a, 001 b b, 002 a a, 002 a b,}}\\{\rm{002 b a, 002 b b, 010 a a, 010 a b, 010 b a, 010 b b, 011 a b, 011 b a, 011 b b, 012 a b,}}\\{\rm{012 b a, 012 b b, 020 a b, 020 b a, 020 b b, 021 a b, 021 b a, 021 b b, 022 a b, 022 b a, }}\\{\rm{022 b b, 100 a b, 100 b a, 100 b b, 101 a b, 101 b a, 101 b b, 102 a b, 102 b a, 102 b b, }}\\{\rm{110 a b, 110 b a, 110 b b, 111 a b, 111 b a, 111 b b, 112 a b, 112 b a, 112 b b, 120 a b,}}\\{\rm{120 b a, 120 b b, 121 a b, 121 b a, 121 b b, 122 a b, 122 b a, 122 b b, 200 a b, 200 b a,}}\\{\rm{200 b b, 201 a b, 201 b a, 201 b b, 202 a b, 202 b a, 202 b b, 210 a b, 210 b a, 210 b b, }}\\{\rm{211 a b, 211 b a, 211 b b, 212 a b, 212 b a, 212 b b, 220 a b, 220 b a, 220 b b, 221 a b,}}\\{\rm{221 b a, 221 b b, 222 a b, 222 b a, 222 b b,}}\end{array}\)

Step-by-step solution

Definition of Concept

Permutations: A permutation of a set is a loosely defined arrangement of its members into a sequence or linear order, or, if the set is already ordered, a rearrangement of its elements, in mathematics. The act of changing the linear order of an ordered set is also referred to as "permutation."

Lexicographic order: The lexicographic or lexicographical order (also known as lexical order or dictionary order) in mathematics is a generalization of the alphabetical order of dictionaries to sequences of ordered symbols or, more broadly, elements of a totally ordered set.

List the names of the files in lexicographic order where we order letters using the usual alphabetic order of letters

Considering the given information:

Letter = a or b

Digit =\({\rm{0,1,2}}\)

Using the following concept:

Letters are ordered alphabetically in lexicographic order.\({{\rm{a}}_{\rm{i}}}{\rm{ < }}{{\rm{b}}_{\rm{i}}}\)For the first i in which\({{\rm{a}}_{\rm{i}}}\) and\({{\rm{b}}_{\rm{i}}}\) differ.

The total number of files in the list is:

\(\begin{array}{l}{\rm{ = }}{{\rm{3}}^{\rm{2}}}{\rm{ \times }}{{\rm{2}}^{\rm{2}}}\\{\rm{ = 27 \times 4}}\\{\rm{ = 108}}\end{array}\)

Therefore, the required list in lexicographic order is:

\(\begin{array}{l}{\rm{000 a a, 000 a b, 000 b a, 000 b b, 001 a a, 001 a b, 001 b a, 001 b b, 002 a a, 002 a b,}}\\{\rm{002 b a, 002 b b, 010 a a, 010 a b, 010 b a, 010 b b, 011 a b, 011 b a, 011 b b, 012 a b,}}\\{\rm{012 b a, 012 b b, 020 a b, 020 b a, 020 b b, 021 a b, 021 b a, 021 b b, 022 a b, 022 b a, }}\\{\rm{022 b b, 100 a b, 100 b a, 100 b b, 101 a b, 101 b a, 101 b b, 102 a b, 102 b a, 102 b b, }}\\{\rm{110 a b, 110 b a, 110 b b, 111 a b, 111 b a, 111 b b, 112 a b, 112 b a, 112 b b, 120 a b,}}\\{\rm{120 b a, 120 b b, 121 a b, 121 b a, 121 b b, 122 a b, 122 b a, 122 b b, 200 a b, 200 b a,}}\\{\rm{200 b b, 201 a b, 201 b a, 201 b b, 202 a b, 202 b a, 202 b b, 210 a b, 210 b a, 210 b b, }}\\{\rm{211 a b, 211 b a, 211 b b, 212 a b, 212 b a, 212 b b, 220 a b, 220 b a, 220 b b, 221 a b,}}\\{\rm{221 b a, 221 b b, 222 a b, 222 b a, 222 b b,}}\end{array}\)