AGES 14  - 18

Counting and Probability

To build on the concepts from the Introduction level course, students will build on their Counting and Probability knowledge. A thorough introduction for students in grades 7-10 to counting and probability topics such as permutations, combinations, Pascal's triangle, geometric probability, basic combinatorial identities, the Binomial Theorem, and more. Pigeonhole Principle Conditional Probability Graph Theory

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Who is this course for
This course is for students who have completed our Introduction to Counting and Probability course.
This course typically takes 3-6 months to complete. This depends on the student's experience and how fast they can master the concepts and the knowledge. Our instructors move at the pace of the student, it may take extra time for some students to reinforce what they have learned.
Curriculum
Chapter 1: Review of Counting and Probability Basics
Review of basic counting and probability, expected value, Pascal’s Triangle and Binomial Theorem, and the summation notation
Chapter 2: Sets and Logic
Sets, logics, quantifiers
Chapter 3: A Piece of PIE
Principle of Inclusion and Exclusion
Chapter 4: Constructive Counting and 1-1 correspondances
Constructive counting, count by finding1-1 correspondances
Chapter 5: Pigeonhole Principle
Pigeonhole Principle
Chapter 6: Constructive Expectation
Linearity of Expectation
Chapter 7: Distributions
Count the number of ways to place indistinguishable items into distinguishable boxes
Chapter 8: Mathematical Induction
Induction
Chapter 9: Fibonacci Number
Introduction and application for Fibonacci Numbers
Chapter 10: Recursion
Recursion, Catalan Numbers
Chapter 11: Conditional Probability
Conditional Probability, and their applications
Chapter 12: Combinatorial Identities
Strategies of proving Combinatorial Identities
Chapter 13: Events with states
State diagrams, random walks, events with infinite states
Chapter 14: Generating Functions
Definition, and applications in deriving the binomial theorem, as well as formulas for distribution, partitions, and Fibonacci numbers
Chapter 15: Graph Theory
Definitions, Basic Properties, Cycles and Paths, Planar Graphs, Eulerian and Hamiltonian paths
Chapter 16: Challenge Problems
Hard problems in counting that use combinations of topics above
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