Question

Place these permutations of \({\rm{\{ 1,2,3,4,5,6\} }}\) in lexicographic order: \({\rm{234561,231456,165432,156423,543216,541236,231465,314562,432561,654321,654312,435612}}{\rm{.}}\)

Short answer

The required lexicographic order is:

\(\begin{array}{*{20}{l}}{{\rm{156423,165432,231456,231465,234561,31562,432561,435612,541236,541236,}}}&{\rm{ }}\\{{\rm{543216,654312,654321}}{\rm{.}}}&{}\end{array}\)

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.

Find the given number in lexicographic order

Considering the given information:

Numbers are:

\(\begin{array}{l}{\rm{234561,231456,165432,156423,543216,541236,231465,314562,432561,}}\\{\rm{654321,654321,435612}}\end{array}\)

Using the following concept:

Order of lexicography:

A sequence of numerical digits is used to represent N on a negative integer in lexicographic order.

\(\begin{array}{l}{\rm{156423 < 165432 < 231456 < 231465 < 234561 < 31562 < 432561 < 435612 < 541236 < 541236}}\\{\rm{ < 543216 < 654312 < 654321}}\end{array}\)

Since the first string appears in the lexicographic order, the second and so on are shown as:

\(\begin{array}{*{20}{l}}{{\rm{156423,165432,231456,231465,234561,31562,432561,435612,541236,541236,}}}&{\rm{ }}\\{{\rm{543216,654312,654321}}}&{}\end{array}\)

Therefore, the required lexicographic order is:

\(\begin{array}{*{20}{l}}{{\rm{156423,165432,231456,231465,234561,31562,432561,435612,541236,541236,}}}&{\rm{ }}\\{{\rm{543216,654312,654321}}{\rm{.}}}&{}\end{array}\)