Question

Place these permutations of \({\rm{\{ 1,2,3,4,5\} }}\) in lexicographic order: \({\rm{43521,15432,45321,23451,23514,14532,21345,45213,31452,31542}}\).

Short answer

The required lexicographic order is:

\({\rm{14532,15432,21345,23451,23514,31452,31542,43521,45213,45321}}\)

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 generalisation 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\({\rm{43512,15432,45321,23451,23514,14532,21345,45231,31452,31542}}\).

Using the following concept:

Order of lexicography:

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

\({\rm{14532 < 15432 < 21345 < 23451 < 23514 < 31452 < 31542 < 43521 < 45213 < 45321}}{\rm{.}}\)

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

\({\rm{14532,15432,21345,23451,23514,31452,31542,43521,45213,45321}}{\rm{.}}\)

Therefore, the required lexicographic order is:

\({\rm{14532,15432,21345,23451,23514,31452,31542,43521,45213,45321}}\).