Question

(a) Construct the permutation numbered 39 in the dictionary ordering of the permutations of the letters \\{1,2,3,4,5\\} . Remember this is actually the 40 th permutation, since the numbering of permutations starts with \(0 .\) (b) Construct the permutation numbered 387 in the permutations of \(\\{1,2,3,4,5,6\\} .\) (c) Construct the permutation numbered 3764 in the permutations of \(\\{1,2,3,4,5, 6,7\\}\). (d) Construct the permutation numbered 27.459 in the permutations of \(\\{1,2,3,4,5,6,7,8\\}\).

Short answer

(a) [2, 5, 1, 3, 4]; (b) [4, 2, 6, 1, 5, 3]; (c) [6, 1, 5, 7, 3, 4, 2]; (d) [6, 8, 4, 2, 7, 1, 3, 5].

Step-by-step solution

Understand the Problem

The goal is to find specific permutations from a set of numbers in their lexicographic order. Given numbers are based on their zero-indexed permutation order, but the question references a one-indexed order, so we must find positions respectively.

Number of Permutations Calculation

For a set of size \( n \), there are \( n! \) total permutations. This allows us to determine the size required to reach a specific permutation order.

Adjust Index Reference

Translate the question's index to zero-based by subtracting 1. This means for (a), we need the 39th permutation (since it's 0-indexed). For (b) find the permutation at index 386.

Find 39th Permutation of \{1,2,3,4,5\}

Use the factorial system to determine positions, with 5 elements, \( 4! = 24 \). Calculate \( 39 \div 24 = 1 \) and use the ceiling for the first digit is '2'. Next, \( 39 \mod 24 = 15 \), repeat for the remaining digits, using \( 3! = 6 \) etc., giving the permutation [2, 5, 1, 3, 4].

Find 387th Permutation of \{1,2,3,4,5,6\}

Perform similar step-by-step factorial division with \( 6! = 720 \), \( 387 \div 120 = 3 \). Starting with '4', use modulus for next positions giving [4, 2, 6, 1, 5, 3].

Find 3764th Permutation of \{1,2,3,4,5,6,7\}

Apply the method on 7 elements with \( 7! = 5040 \). Divide 3763 by 720: quotient 5, pointing initial digit, and obtain the sequence to [6, 1, 5, 7, 3, 4, 2].

Find 27,459th Permutation of \{1,2,3,4,5,6,7,8\}

For 8 elements \( 8! = 40320 \), continuing: 27,458 divided by 5040 gives 5 (selecting 6), and iteratively arrives at [6, 8, 4, 2, 7, 1, 3, 5].

Key concepts

Lexicographic Order

In mathematics, lexicographic order (or dictionary order) is a way to arrange sequences. Think of it as a way words are alphabetically sorted in a dictionary. When applied to numbers, it arranges them similarly based on their integer sequence. For permutations, this involves arranging every possible order of given elements.
For example, with the set \( \{1, 2, 3\} \), lexicographic order for permutations would list the permutations as \( [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1] \).
This approach is especially useful when needing to determine the "nth" permutation in a sequence. The permutation number is derived by systematically using factorial math to switch through positions, like flipping through a dictionary up to a certain word.

Factorial Calculation

Factorial calculation, represented as \( n! \), involves multiplying a series of descending natural numbers. It's critical in permutations because it indicates the total number of ways to arrange \( n \) distinct items.
For instance, \( 4! \) (read as "four factorial") equals \( 4 \times 3 \times 2 \times 1 = 24 \), showing there are 24 different ways to order a set of four items.
In permutation calculations, factorials help determine positions by dividing the sequence range, systematically narrowing down to precise elements. This approach dissect complex problems into manageable factorial calculations, guiding through step-by-step division and remainder processes.

Zero-Based Indexing

Zero-based indexing is a system where counting starts at zero rather than one. In computer science and mathematics, this approach is common, and functions as a foundation for array and sequence processing.
For permutation problems, this means the first element in a sequence has an index of 0. Therefore, when tasked with finding a permutation like the 39th permutation, mathematically, we reference the position as 38 in calculations.
By translating given one-index references to zero-based contexts, the solution more naturally fits many algorithm structures. Subtracting one from given indices simplifies this transition, ensuring alignment with computational conventions.

Combinatorics

Combinatorics is the study of counting, arranging, and understanding how items within a set can be selected in specific ways. It intertwined deeply with permutations, which are specific types of arrangements that consider element order.
This area of mathematics answers questions like, "How many ways can a set be rearranged?" or "What is the nth arrangement in a lexicographic order?"
Within combinatorics, permutations utilize formulas and logical steps like factorial operations and index reasoning to solve complex arrangement puzzles. Each concept helps break down permutations—like identifying specific orders efficiently in tasks such as those involving dictionaries of permutations.