• IB
  • IB Docs (2) Team
    Logout
  • Maths
  • Biology
  • Chemistry
  • Physics
  • Combined Science
  • English Language
  • Geography
  • Other Subjects
GCSE Maths
Edexcel Topic QuestionsRevision NotesPast PapersPast Papers Questions
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast Papers
GCSE Biology
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Gateway Topic QuestionsRevision NotesPast Papers
GCSE Chemistry
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Gateway Topic QuestionsRevision NotesPast Papers
GCSE Physics
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Gateway Topic QuestionsRevision NotesPast Papers
GCSE Combined Science
Edexcel Combined: Biology Topic QuestionsRevision NotesPast Papers
Edexcel Combined: Chemistry Topic QuestionsRevision NotesPast Papers
Edexcel Combined: Physics Revision NotesPast Papers
AQA Combined: Biology Topic QuestionsRevision NotesPast Papers
AQA Combined: Chemistry Topic QuestionsRevision NotesPast Papers
AQA Combined: Physics Topic QuestionsRevision NotesPast Papers
OCR Gateway Combined: Biology Topic QuestionsRevision Notes
OCR Gateway Combined: Chemistry Revision Notes
OCR Gateway Combined: Physics Revision Notes
GCSE English Language
AQA Revision NotesPractice PapersPast Papers
Edexcel Past Papers
OCR Past Papers
GCSE Geography
AQA Topic QuestionsRevision Notes
Edexcel Topic Questions
GCSE Other Subjects
AQA English LiteratureBusinessComputer ScienceEconomicsFurther MathsGeographyHistoryPsychologySociologyStatistics
Edexcel English LiteratureBusinessComputer ScienceGeographyHistoryPsychologyStatistics
OCR English LiteratureBusinessComputer ScienceEconomicsPsychology
OCR Gateway GeographyHistory
  • Maths
  • Biology
  • Chemistry
  • Physics
  • Double Science
  • Economics
  • English Language
  • Geography
  • Other Subjects
IGCSE Maths
Edexcel Topic QuestionsRevision NotesPast PapersBronze-Silver-Gold Questions
CIE (Extended) Topic QuestionsRevision NotesPast Papers
CIE (Core) Topic QuestionsPast Papers
IGCSE Biology
Edexcel Topic QuestionsRevision NotesPast Papers
CIE 2020-2022 Topic QuestionsRevision NotesPast Papers
CIE 2023-2025 Topic QuestionsRevision NotesPast Papers
IGCSE Chemistry
Edexcel Topic QuestionsRevision NotesPast Papers
CIE 2020-2022 Topic QuestionsRevision NotesPast Papers
CIE 2023-2025 Topic QuestionsRevision NotesPast Papers
IGCSE Physics
Edexcel Topic QuestionsRevision NotesPast Papers
CIE 2020-2022 Topic QuestionsRevision NotesPast Papers
CIE 2023-2025 Topic QuestionsRevision NotesPast Papers
IGCSE Double Science
Edexcel Double: Biology Topic QuestionsRevision NotesPast Papers
Edexcel Double: Chemistry Topic QuestionsRevision NotesPast Papers
Edexcel Double: Physics Topic QuestionsRevision NotesPast Papers
IGCSE Economics
CIE Topic QuestionsRevision NotesPast Papers
IGCSE English Language
CIE Revision NotesPractice PapersPast Papers
Edexcel Past Papers
IGCSE Geography
CIE Revision NotesTopic QuestionsPast Papers
Edexcel Topic QuestionsRevision NotesPast Papers
IGCSE Other Subjects
CIE Additional MathsEnglish LiteratureBusinessComputer ScienceHistorySociology
Edexcel English LiteratureBusinessComputer ScienceHistoryFurther Maths
  • Maths
  • Biology
  • Chemistry
  • Physics
  • English Language
  • Other Subjects
AS Maths
Edexcel Pure MathsMechanicsStatistics
AQA Pure MathsMechanicsStatistics
OCR Pure MathsMechanicsStatistics
CIE Pure 1Pure 2MechanicsProbability & Statistics 1
Edexcel IAS Pure 1Pure 2MechanicsStatistics
AS Biology
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast Papers
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Revision Notes
AS Chemistry
Edexcel Revision Notes
AQA Topic QuestionsRevision NotesPast Papers
OCR Revision Notes
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Revision Notes
AS Physics
Edexcel Revision Notes
AQA Topic QuestionsRevision NotesPast Papers
OCR Revision NotesPast Papers
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Revision Notes
AS English Language
AQA Past Papers
Edexcel Past Papers
OCR Past Papers
AS Other Subjects
AQA BusinessComputer ScienceEconomicsEnglish LiteratureFurther MathsGeographyHistoryPsychologySociology
Edexcel BusinessEconomicsEnglish LiteratureFurther MathsGeographyHistoryPsychology
OCR BusinessComputer ScienceEconomicsEnglish LiteratureFurther Maths AGeographyHistoryPsychologySociology
CIE Further Maths
  • Maths
  • Biology
  • Chemistry
  • Physics
  • English Language
  • Economics
  • Further Maths
  • Psychology
  • Other Subjects
A Level Maths
Edexcel Pure MathsMechanicsStatistics
AQA Pure MathsMechanicsStatistics
OCR Pure MathsMechanicsStatistics
CIE Pure 1Pure 3MechanicsProbability & Statistics 1Probability & Statistics 2
Edexcel IAL Pure 1Pure 2Pure 3Pure 4Mechanics 1Mechanics 2Statistics 1Statistics 2Decision 1
A Level Biology
Edexcel Topic QuestionsPast Papers
Edexcel A (SNAB) Revision Notes
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast PapersGold Questions
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Topic QuestionsRevision NotesPast Papers
A Level Chemistry
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast PapersGold Questions
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Topic QuestionsRevision NotesPast Papers
A Level Physics
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast Papers
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Topic QuestionsRevision NotesPast Papers
A Level English Language
AQA Past Papers
CIE Past Papers
Edexcel Past Papers
OCR Past Papers
Edexcel IAL Past Papers
A Level Economics
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsPast Papers
OCR Past Papers
CIE Past Papers
A Level Further Maths
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Past Papers
OCR Past Papers
CIE Past Papers
Edexcel IAL Past Papers
A Level Psychology
AQA Topic QuestionsRevision NotesPast Papers
CIE Past Papers
Edexcel Past Papers
OCR Past Papers
Edexcel IAL Past Papers
A Level Other Subjects
AQA BusinessComputer ScienceEconomicsEnglish LiteratureGeographyHistorySociology
CIE BusinessComputer ScienceEconomicsEnglish LiteratureGeographySociology
Edexcel BusinessEconomics AEnglish LiteratureGeographyHistory
OCR BusinessComputer ScienceEconomicsEnglish LiteratureGeographyHistorySociology
Edexcel IAL English LiteratureGeography
CIE IAL History
  • Biology
  • Chemistry
  • Physics
  • Other Subjects
O Level Biology
CIE Topic QuestionsPast Papers
O Level Chemistry
CIE Topic QuestionsPast Papers
O Level Physics
CIE Topic QuestionsPast Papers
O Level Other Subjects
CIE Additional MathsMaths D
  • Maths
  • Biology
  • Chemistry
  • Physics
Pre U Maths
CIE Topic QuestionsPast Papers
Pre U Biology
CIE Topic QuestionsPast Papers
Pre U Chemistry
CIE Topic QuestionsPast Papers
Pre U Physics
CIE Topic QuestionsPast Papers
  • Maths
  • Biology
  • Chemistry
  • Physics
  • Economics
IB Maths
Maths: AA HL Topic QuestionsRevision NotesPractice Papers
Maths: AI HL Topic QuestionsRevision NotesPractice Papers
Maths: AA SL Topic QuestionsRevision NotesPractice Papers
Maths: AI SL Topic QuestionsRevision NotesPractice Papers
IB Biology
Biology: SL Topic QuestionsRevision NotesPractice Papers
Biology: HL Topic QuestionsRevision NotesPractice Papers
IB Chemistry
Chemistry: SL Topic QuestionsRevision NotesPractice Papers
Chemistry: HL Topic QuestionsRevision NotesPractice Papers
IB Physics
Physics: SL Topic QuestionsRevision NotesPractice Papers
Physics: HL Topic QuestionsRevision NotesPractice Papers
IB Economics
Economics: SL Revision Notes

DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 1. Number & Algebra / 1.7 Permutations & Combinations / 1.7.1 Counting Principles


1.7.1 Counting Principles


Counting Principles

What is meant by counting principles?

  • The fundamental counting principle states that if there are m ways to do one thing and n ways to do another there are m × n ways to do both things
  • Applying counting principles allows us to…
    • … analyse patterns and make generalisations about real work situations
    • … find the number of permutations of n items
    • … find the number of ways of choosing an item from a list of n items
    • … find the number of ways of choosing r items from n items
    • … find the number of ways of permutating r items from n items
  • The topic of counting principles is a particularly interesting part of mathematics that can lead to the development of working with very large numbers
  • It is always vital to consider whether objects taken from each list can be repeated or not
    • For example a four digit PIN from ten numbers where each number can be used repeatedly would be 10 × 10 × 10 × 10
      • There are 10 options for the first and ten options for the second number and so on
    • If the numbers could only be used once then the number of options for each digit would reduce with each digit
      • There are 10 options for the first, nine options for the second, eight for the third and so on
      • This concept will be explored further in the permutations revision note

How do I choose an item from a list of m items AND another item from a list of n items?

  • If a question requires you to choose an item from one list AND an item from another list you should multiply the number of options in each list
    • In general if you see the word 'AND' you will most likely need to 'MULTIPLY'
  • For example if you are choosing a pen and a pencil from 4 pens and 5 pencils:
    • You can choose 1 item from 4 pens AND 1 item from 5 pencils
    • You will have 4 × 5 different options to choose from

 

How do I choose an item from a list of m items OR another item from a list of n items?

  • If a question requires you to choose an item from one list OR an item from another list you should add the number of options in each list
    • In general if you see the word 'OR' you will most likely need to 'ADD'
  • For example if you are choosing a pen or a pencil from 4 pens and 5 pencils:
    • You can choose 1 item from 4 pens OR 1 item from 5 pencils
    • You will have 4 + 5 different options to choose from

Exam Tip

  • Counting principles and factorials are tightly interlinked with permutations and combinations
  • Make sure you fully understand the concepts in this revision note as they will be fundamental to answering perms and combs exam questions

Worked Example

Harry is going to a formal event and is choosing what accessories to add to his outfit. He has seven different ties, four different bow ties and five different pairs of cufflinks. How many different ways can Harry get ready if he chooses:

a)
Either a tie, a bow tie or a pair of cufflinks?

1-7-1-ib-aa-hl-counting-principles-we-solution-i

b)
A pair of cufflinks and either a tie or a bow tie?

1-7-1-ib-aa-hl-counting-principles-we-solution-ii



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
          • 1.2 Exponentials & Logs
            • 1.2.1 Introduction to Logarithms
              • 1.2.2 Laws of Logarithms
                • 1.2.3 Solving Exponential Equations
                • 1.3 Sequences & Series
                  • 1.3.1 Language of Sequences & Series
                    • 1.3.2 Arithmetic Sequences & Series
                      • 1.3.3 Geometric Sequences & Series
                        • 1.3.4 Applications of Sequences & Series
                          • 1.3.5 Compound Interest & Depreciation
                          • 1.4 Simple Proof & Reasoning
                            • 1.4.1 Proof
                            • 1.5 Further Proof & Reasoning
                              • 1.5.1 Proof by Induction
                                • 1.5.2 Proof by Contradiction
                                • 1.6 Binomial Theorem
                                  • 1.6.1 Binomial Theorem
                                    • 1.6.2 Extension of The Binomial Theorem
                                    • 1.7 Permutations & Combinations
                                      • 1.7.1 Counting Principles
                                        • 1.7.2 Permutations & Combinations
                                        • 1.8 Complex Numbers
                                          • 1.8.1 Intro to Complex Numbers
                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
                                                • 1.9.1 Geometry of Complex Numbers
                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 2. Functions
                                                            • 2.1 Linear Functions & Graphs
                                                              • 2.1.1 Equations of a Straight Line
                                                              • 2.2 Quadratic Functions & Graphs
                                                                • 2.2.1 Quadratic Functions
                                                                  • 2.2.2 Factorising & Completing the Square
                                                                    • 2.2.3 Solving Quadratics
                                                                      • 2.2.4 Quadratic Inequalities
                                                                        • 2.2.5 Discriminants
                                                                        • 2.3 Functions Toolkit
                                                                          • 2.3.1 Language of Functions
                                                                            • 2.3.2 Composite & Inverse Functions
                                                                              • 2.3.3 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
                                                                                                                                                                                                  • 4.1.2 Statistical Measures
                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                                                                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Amber Perry

                                                                                                                                                                                                                                                                                                        Author: Amber

                                                                                                                                                                                                                                                                                                        Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.


                                                                                                                                                                                                                                                                                                        Save My Exams Logo
                                                                                                                                                                                                                                                                                                        Resources
                                                                                                                                                                                                                                                                                                        Home Join Support

                                                                                                                                                                                                                                                                                                        Members
                                                                                                                                                                                                                                                                                                        Members Home Account Logout

                                                                                                                                                                                                                                                                                                        Company
                                                                                                                                                                                                                                                                                                        About Us Contact Us Jobs Terms Privacy Facebook Twitter

                                                                                                                                                                                                                                                                                                        Quick Links
                                                                                                                                                                                                                                                                                                        GCSE Revision Notes IGCSE Revision Notes A Level Revision Notes Biology Chemistry Physics Maths 2022 Advance Information

                                                                                                                                                                                                                                                                                                         
                                                                                                                                                                                                                                                                                                        © IB Documents (2) Team & u/aimlesskr
                                                                                                                                                                                                                                                                                                        IBO was not involved in the production of, and does not endorse, the resources created by Save My Exams.