# AoPS Counting & Probability Videos

The videos listed below go with our *Introduction to Counting & Probability* textbook. The videos are grouped by the corresponding chapter of the textbook.

- Chapter 1: Counting is Arithmetic
- Chapter 2: Basic Counting Techniques
- Chapter 3: Correcting for Overcounting
- Chapter 4: Committees and Combinations
- Chapter 5: More with Combinations
- Chapter 6: Some Harder Counting Problems
- Chapter 7: Introduction to Probability
- Chapter 8: Basic Probability Techniques
- Chapter 9: Think About It!
- Chapter 10: Geometric Probability
- Chapter 11: Expected Value
- Chapter 12: Pascal's Triangle
- Chapter 13: The Hockey Stick Identity
- Chapter 14: The Binomial Theorem

Chapter 1: Counting is Arithmetic | back to top |

Counting numbers in lists in which the each number is 1 more than the previous number.

Counting lists that change by a number besides 1.

Counting with 2-circle Venn diagrams, and with subtraction.

Using a 3-circle Venn diagram to solve a problem, and introducing the Principle of Inclusion-Exclusion.

Using multiplication to count.

Using multiplication to count the number of different kinds of day I can have.

Introducing factorials

Computing factorials and manipulating factorials to evaluate expressions involving them.

Introducing counting permutations.

Learning some notation for permutations, and how to write them in terms of factorials.

Chapter 2: Basic Counting Techniques | back to top |

An introduction to using casework in counting problems.

Two more casework counting problems.

We introduce the tactic of counting the complement; that is, counting what you don't want.

Nasty casework? Try complementary counting.

Introducing a constructive method for counting.

Solving counting problems with restrictions by dealing with the restrictions first.

Solving another counting problem with restrictions by dealing with the restrictions first.

Chapter 3: Correcting for Overcounting | back to top |

Using division to correct for overcounting.

Counting the number of possible arrangements of letters in words with repeated letters.

Correcting overcounting with division to tackle challenging problems involving diagonals of polygons and polyhedra.

Counting the number of ways to seat people at a round table.

Counting the number of ways to place keys on a keychain.

Chapter 4: Committees and Combinations | back to top |

Introduction to combinations.

Another introductory counting with combinations problem.

Learn how to compute combinations.

Find a formula for C(n,r) in terms of factorials.

Why do we define 0! to be 1?

We introduce and prove the identity C(n,r) = C(n,n-r) in two ways.

Chapter 5: More with Combinations | back to top |

Counting paths on a grid with combinations.

Applying combinations to a counting problem at the San Diego zoo.

Solving a combination problem with restrictions in two different ways.

Tackling a tough combinations with restrictions problem.

Learning the importance of "with replacement" and "without replacement" in counting problems.

Learning the importance of distinguishability in counting problems.

Tackling more examples of the importance of distinguishability in counting problems.

Chapter 6: Some Harder Counting Problems | back to top |

Count the zeroes at the end of 320!

Taking a constructive approach to counting the number of regions into which we can cut a plane with some number of lines (and a circle!)

Tackling a challenging counting problem that has many restrictions.

Learning the importance of "with replacement" and "without replacement" in counting problems.

Chapter 7: Introduction to Probability | back to top |

Learning why it's important that we count equally likely outcomes when doing probability problems.

Tackling probability problems that involve combinations.

Learning why we must be consistent with our approaches when we solve probability problems by counting successful outcomes and counting possible outcomes.

Correcting a common probability misconception involving problems

Tackling a challenging probability problem as two counting problems.

Finding the probability I come in second in the National MATHCOUNTS Countdown round.

Chapter 8: Basic Probability Techniques | back to top |

Finding probabilities using casework.

Finding the probability you want by first finding the probability of what you don't want first.

Finding the probability of what we don't want in order to evaluate a probability we want.

Finding a desired probability that an event occurs by first finding the probability that the event does *not* occur.

Solving a harder geyser probability problem with complementary probability.

Finding the probability you are psychic, or un-psychic, by multiplying the probabilities of independent events.

Finding the probability that the best football weekend ever occurs.

Learning the importance of "independent" in multiplying the probabilities of "independent" events.

Finding the probability a repeated event occurs a certain number of times out of a given number of trials.

Chapter 9: Think About It! | back to top |

Solving a complicated probability problem by realizing it isn't so complicated, after all.

Finding a slick solution to a probability problem by just . . . thinking about it!

Making a probability problem easier by just thinking about it in a different way.

Chapter 10: Geometric Probability | back to top |

Using geometry to solve probability problems.

Using area to solve probability problems.

Using area to solve a challenging probability problem.

Chapter 11: Expected Value | back to top |

Learning about expected value by playing some games.

Playing another expected value game.

Using expected value to determine how a game is rigged.

Discovering an expected value paradox while flipping a coin.

Discovering an expected value "paradox" involving money in envelopes.

Chapter 12: Pascal's Triangle | back to top |

Discovering Pascal's Triangle.

Learning Pascal's Identity.

Discovering patterns in Pascal's Triangle.

Chapter 13: The Hockey Stick Identity | back to top |

Introducing a problem we'll use to learn about the Hockey Stick Identity.

Continuing our exploration of the Hockey Stick Identity by counting how many ways we can distribute money to three people.

Finding the number of ways to distribute dollar bills among four people.

We find the Hockey Stick in the Hockey Stick Identity.

Deriving the Hockey Stick Identity

Chapter 14: The Binomial Theorem | back to top |

Discovering the Binomial Theorem.

Finding a formula for raising a binomial to a positive integer power, as well as finding an algebraic explanation for why the Binomial Theorem is true.

Finding a counting proof for the Binomial Theorem.

Using the Binomial Theorem to expand powers of binomials.

Using the Binomial Theorem to expand the fourth power of a binomial.

Using the Binomial Theorem to find specific terms in expansions of powers of binomials.

Finding, and proving, a cool pattern in Pascal's Triangle.

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