Curriculum · Counting & Probability
Counting & Probability
From the product rule to expected value — combinatorial reasoning, the most learnable superpower in competition math.
The Art of Counting
Counting Fundamentals
The product rule, the sum rule, and the discipline of counting without double-counting.
- Apply product and sum rules correctly
- Detect and correct double counting
- Count by building objects step by step
Permutations & Combinations
Arrangements, selections, and the bridge between them — with repeated elements handled honestly.
- Choose between nPr and nCr
- Handle repeated objects
- Count circular arrangements
prerequisite: counting-fundamentals
Casework & Complementary Counting
Splitting problems into clean cases — and the liberating move of counting what you don't want.
- Partition into exhaustive, disjoint cases
- Recognize when complementary counting wins
- Combine casework with the product rule
prerequisite: permutations-combinations
The Binomial Theorem
Pascal's triangle, binomial identities, and coefficient extraction.
- Expand with the binomial theorem
- Prove identities combinatorially
- Extract coefficients
prerequisite: permutations-combinations
Probability & Expectation
Probability Foundations
Sample spaces, equally likely outcomes, independence, and conditional probability.
- Build sample spaces explicitly
- Multiply along independent events
- Compute conditional probabilities
prerequisite: counting-fundamentals
Expected Value & Linearity
The most elegant tool in probability: linearity of expectation, indicators, and why it works even with dependence.
- Compute expected value from definitions
- Apply linearity of expectation
- Design indicator variables
prerequisite: probability-foundations
The Pigeonhole Principle
Existence by counting: if there are more pigeons than holes, two share.
- State and apply pigeonhole
- Choose the right holes
- Use the generalized version
prerequisite: counting-fundamentals
Geometric Probability
Probability as area — continuous sample spaces and clever regions.
- Model with regions
- Compute area ratios
- Handle time-interval meeting problems
prerequisite: probability-foundations, coordinate-plane-basics
Counting with Recursion
States, transitions, and the recurrences behind tiling and path problems.
- Define states and transitions
- Set up recurrences
- Compute by building up
prerequisite: casework-complementary