Binomial Theorem

Explore the proofs and applications of the Binomial Theorem in this comprehensive 8-hour video playlist. Delve into binomial expansions and their ubiquitous presence in mathematics, from basic algebra to permutations and combinations. Discover the fascinating connection between factorials and Pascal's Triangle, and learn to solve complex problems related to polynomials and probability. Cover topics such as basic binomial expansions, Pascal's Triangle, binomial identities, finite series with binomial coefficients, probability questions, nCr notation, factorial notation, and Sierpinski's Triangle. Work through numerous examples and exam-style questions to reinforce your understanding of this fundamental mathematical concept.

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.