This course develops the Binomial Theorem through expansions, factorial and nCr notation, Pascal’s Triangle, coefficient calculations, and proofs of binomial identities. It also applies these ideas to finite series, permutations and combinations, polynomial expressions, and probability.

Syllabus

Prologue to Binomial Theorem (Arrangements).
Basic Binomial Expansions.
Binomial Expansion: Simple Examples.
Binomial Theorem & Pascal's Triangle.
Binomial Theorem: Introductory Exercises.
Tricky Permutations & Combinations Question.
Proving Binomial Identities.
Finite Series w/ Binomial Coefficients: Substitution.
Finite Series w/ Binomial Coefficients: Differentiation.
Finite Series w/ Binomial Coefficients: Integration (1 of 2).
Finite Series w/ Binomial Coefficients: Integration (2 of 2).
Binomial Identities: Pascal Example.
Greatest Coefficient in Binomial Expansion.
Evaluating Specific Binomial Coefficients.
Evaluating Coefficients From Two Expansions.
Proving Binomial Identities: Comparing Coefficients.
Probability Question: With & Without Binomial.
Pascal's Identity (1 of 2).
Pascal's Identity (2 of 2).
Expressing Terms in a Generalised Binomial Expansion.
The Properties of Pascal's Triangle in nCr Notation.
Introduction to nCr Notation.
How are Binomial Coefficients related to Pascal's Triangle?.
The Weirdness of Pascal's Triangle.
An ingenious & unexpected proof of the Binomial Theorem (1 of 2: Prologue).
An ingenious & unexpected proof of the Binomial Theorem (2 of 2: Proof).
Demonstrating the Binomial Theorem from Pascal's Triangle.
Sierpinski's Triangle.
Factorials.
Determining & Calculating the Greatest Term (example question).
Finding & Evaluating the Greatest Coefficient (example question).
Relating Consecutive Coefficients (and finding the greatest one).
What is the Greatest Coefficient (and why do we care about it)?.
Binomial Identity with Even Coefficients Only?.
Binomial / Permutations & Combinations Proof (1 of 2).
Binomial / Permutations & Combinations Proof (2 of 2).
Recognising & Substituting Part of a Binomial Expansion.
Simplifying Products w/ Factorial Notation.
What power of 2 is a factor of "100!"?.
Identifying the Constant Term in a Binomial Expansion.
Expanding Binomials Involving Substitution.
Separating Rational & Irrational Components in Binomial Expansion.
Stating the Binomial Theorem w/ Factorial Notation.
Developing the Binomial Theorem w/ Pascal's Triangle.
Introduction to Factorial Notation (1 of 2: Why use "!"?).
Introduction to Factorial Notation (2 of 2: How to simplify factorial terms).
Shorthand Notation for nCr.
Approximating a Decimal Expansion with Binomial Theorem.
Why do all rows of Pascal's triangle add to powers of 2?.
Sigma Notation & Properties of Pascal's Triangle.
nCr Notation.
Why Notation & Terminology Are Awesome, Not Boring.
Binomial Expansions - Simple Example.
Binomial Coefficients & Pascal's Triangle.
Evaluating Specific Binomial Coefficients (Exam Questions).
Summing Binomial Coefficients (Exam Question).
Binomial Coefficients Proof Question.