Year 12 Unit 1 Specialist Mathematics Study Notes & Exam Preparation for Solving Permutations

Year 12 Unit 1 Specialist Mathematics (ACMSM001): Permutations (Combinatorics)
Australian Curriculum
Study Notes & Exam Preparation


I. Basic Concepts of Permutations:

1. Definition:

A permutation refers to the arranged sequence of items. In the context of permutations, the
sequence is extremely important. For a set of 'n' distinct objects, there are 'n!' permutations.



2. Factorial Notation:

The factorial of a positive integer 'n,' denoted as 'n!,' is the product of all positive integers from 1 to
'n.'

-For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.



II. Types of Permutations:

1. Linear Permutations:

In linear permutations, objects are arranged in a straight line. The number of linear permutations for
'n' distinct objects is 'n!.'



2. Circular Permutations:

In circular permutations, objects are arranged in a circle. The number of circular permutations for 'n'
distinct objects is '(n-1)!.'



III. Permutations with Restrictions:

1. Permutations of a Subset:

Often, problems involve selecting 'k' elements from 'n' distinct elements and arranging them. Use the
formula: P(n, k) = n! / (n-k)! to calculate permutations of a subset.



2. Permutations with Repeated Elements:

When objects are not all distinct, divide by the factorial of the number of identical objects to avoid
overcounting.