Counting & Probability
Discrete and Data
Discrete mathematics for problem solvers: counting principles, combinations, casework and complements, stars and bars, the binomial theorem, inclusion–exclusion, recursion, and probability through expected value.
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Counting Without Counting
Unit 1 · Fundamental Counting: Permutations & Combinations
0 of 10 units complete
- 1Unit 1Fundamental Counting: Permutations & Combinations6 parts · about 15 min
Counting Without Counting
How many outfits, licence plates, podiums, poker hands and anagrams of MISSISSIPPI? Multiply instead of listing, tell arranging from choosing, and find out why no two shuffled decks in history have ever matched.
- 2Unit 2Counting Strategies: Casework, Complementary & Overcounting5 parts · about 11 min
Counting Strategies
Split into cases, count what you do not want, or overcount on purpose and divide. Build bijections that turn hard counts into easy ones, and count necklaces that can spin.
- 3Unit 3Distributions: Stars & Bars and Integer Solutions5 parts · about 15 min
Stars and Bars
Share 7 sweets among 3 children and turn every sharing into a row of stars and bars. Then add minimums and maximums, hand out different gifts, and meet the partitions that even Ramanujan found hard.
- 4Unit 4Binomial Theorem, Pascal & Combinatorial Identities5 parts · about 16 min
Pascal's Triangle
One triangle of numbers holds the binomial theorem, the powers of 2, the hockey stick, every lattice path, and (if you colour the odd numbers) a fractal. Click your way through all of it.
- 5Unit 5Inclusion–Exclusion, Derangements & Pigeonhole5 parts · about 12 min
Overlaps, Shuffles and Pigeons
Count a union without counting anyone twice, return hats at random until nobody gets their own (the answer involves e!), and prove things exist just because there are more pigeons than holes.
- 6Unit 6Recursion, Catalan Numbers & Generating Functions5 parts · about 14 min
Counting by Recursion
Too many to count directly? Count the small cases and find how each one builds on the last. Climb stairs, tile with dominoes, draw mountain ranges, and meet the Fibonacci and Catalan numbers that keep turning up everywhere.
- 7Unit 7Probability Fundamentals & Geometric Probability5 parts · about 12 min
Chance by Counting
Probability is counting in disguise: favourable over total. Work out flushes and shared birthdays, settle a 17th-century gambling argument, and estimate π by throwing darts at random.
- 8Unit 8Conditional Probability, Independence & Bayes5 parts · about 12 min
Given That…
New information changes probabilities. Draw marbles without replacement on a tree, test for independence, and find out why a positive result on a 99%-accurate test can still leave you probably healthy. Then face Monty Hall.
- 9Unit 9Expected Value, Linearity & Advanced Probability5 parts · about 12 min
Expected Value
The long-run average of a random quantity, and the superpower that comes with it: linearity, which works even when everything depends on everything. Count hat matches, shared birthdays and coin streaks without breaking a sweat.
- 10Unit 10Identities and Techniques, Proved5 parts · about 12 min
Proofs by Counting
The cleanest proofs in combinatorics count one thing in two ways. Prove the handshake lemma, the hockey stick, the sum of squares, Vandermonde and the committee–chair identity, and see why inclusion–exclusion's signs alternate.