Start here · Essential guide
Counting Principles Every Probability Learner Needs
A plain guide to the multiplication principle, factorials, permutations and combinations, with dice, cards and board game examples you can check by hand.
Methods and Tools
Counting and Combinatorics
Counting outcomes carefully so probabilities come out right.
Counting Possible Passwords and Combination Locks
See how the number of possible codes grows with each added character, using bike locks, PINs and passwords as simple multiplication examples.
Derangements and the Hat Check Problem
Why the chance that nobody gets their own hat back settles near 37 percent however many guests there are, built up from small cases you can list by hand.
Factorials Explained with Shuffled Decks
Learn why a 52-card deck has 52 factorial possible orderings, how to compute small factorials, and just how enormous the full number is.
How Many Five Card Poker Hands Are There
Derive the 2,598,960 possible five-card hands from 52 choose 5, then count four of a kind, full houses and other hand types step by step.
Inclusion-Exclusion for Overlapping Events
Learn to avoid double counting when events overlap, with card and dice examples for two sets and a worked three-set case you can verify by hand.
Pascals Triangle and Binomial Coefficients
See how Pascal's triangle is built, why each entry counts combinations, and how a row gives the probabilities of heads in a series of fair coin tosses.
Permutations Versus Combinations: When Order Matters
Learn the difference between arrangements and selections, how to decide whether order matters, and how to convert one count to the other with examples.
The Pigeonhole Principle in Games
How simple counting can guarantee outcomes, such as matching socks, repeated suits in a hand and shared birthdays, with checked examples and clear limits.
The Stars and Bars Method
A clear explanation of stars and bars for splitting identical items into groups, with checked examples including how many ways three dice can total ten.
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