Linear Permutations of Distinguishable Objects

Study Notes

Linear Permutation of Distinguishable Objects

In mathematics, specifically combinatorics, linear permutations refer to the arrangement of distinguishable objects in a specific order. This concept is often applied when considering factors such as factorial notation, arrangements without repetition, arrangements with repetition, and permutations of subsets.

Factorial Notation

Factorial notation, denoted by the symbol !, expresses the product of all positive integers up to a given number. For example, 5! (pronounced "5 factorial") equals 5 × 4 × 3 × 2 × 1, which is equal to 120. This notation is essential when calculating the number of possible linear permutations.

Arrangements Without Repetition

Arrangements without repetition, also known as permutations, are the different ways of arranging a set of objects in a particular order. If n objects are given, then the number of permutations, P(n), can be calculated using the formula P(n) = n!. For example, if we have four distinct objects, there would be 4 × 3 × 2 × 1 = 24 ways of arranging them linearly.

Arrangements With Repetition

Arrangements with repetition, also known as circular permutations, consider repeating elements within a set. In this context, a circular permutation is a permutation followed by some number of rotations. When dealing with repetitions, the term 'cyclic' is sometimes used instead of 'circular'.

Permutations of Subsets

A subset is a collection of elements taken from another collection. If A is a set with m elements, B a subset of A with k elements, and C the complement of B in A, then the number of linear permutations of the sets A, B, and C can be found by multiplying the number of permutations of each set together: P(B) * P(C).

For instance, let's say we have three unique objects, each represented by a different letter. We want to find all possible linear permutations of these objects. Since there are 3! (6) possible linear permutations of these objects, we can simply list them out: ABC, ACAB, ACBA, ACB, BAC, BAAC, BAACB, BCAA, CB, CBA, CA, CA, CA, and CCA.

Description

Explore the concept of linear permutations in combinatorics, where distinguishable objects are arranged in a specific order. Learn about factorial notation, arrangements without repetition, arrangements with repetition, and permutations of subsets through examples and formulas.