🧩 Problem Types

A comprehensive catalog of common AMC counting and probability problem patterns with recognition cues and solution templates.

📚 Problem Type Overview

🔢 Basic Counting Patterns

Seating & Restrictions - Adjacency and spacing problems
Word Rearrangements - Letter arrangements with constraints
Committees & Conditions - Selection with roles and constraints
Balls in Bins - Integer solutions and distribution problems

🎯 Advanced Counting Patterns

At Least/At Most - Complement and inclusion-exclusion strategies
Grid Paths - Lattice walks with restrictions
Circular Permutations - Ring arrangements and reflections
Derangements - No element in original position

🎲 Probability Patterns

Urn, Coin & String Problems - Replacement vs. no-replacement
Expected Counts - Indicator variables and linearity
Conditional Chains - Total probability and Bayes
Symmetry Probability - Random choices via symmetry

🔍 Specialized Patterns

Hypergeo vs Binomial - Distribution recognition
Digit Counting - Divisibility and base representations

🎯 How to Use This Section

Recognition: Learn the cues that identify each problem type
Template: Follow the solution template for each type
Practice: Work through the AMC-style examples
Pitfalls: Study common mistakes and how to avoid them

📊 Problem Type Matrix

Problem Type	AMC 10	AMC 12	Key Technique
Seating & Restrictions	✅ Core	✅ Core	Adjacency method
Word Rearrangements	✅ Core	✅ Core	Permutations with repetition
Committees & Conditions	✅ Core	✅ Core	Combinations with constraints
Balls in Bins	⚠️ Stretch	✅ Often	Stars and bars
At Least/At Most	⚠️ Stretch	✅ Often	Complement counting
Grid Paths	✅ Often	✅ Core	Lattice path counting
Circular Permutations	⚠️ Stretch	✅ Often	$(n-1)!$ formula
Derangements	❌ No	✅ Stretch	Inclusion-exclusion
Expected Counts	❌ No	✅ Often	Linearity of expectation
Conditional Chains	⚠️ Stretch	✅ Often	Total probability

🏆 Contest Strategy

AMC 10: Focus on basic counting patterns and simple probability
AMC 12: Master all patterns, especially advanced counting and probability
Recognition: Practice identifying problem types quickly
Templates: Memorize solution approaches for common types

Start with Seating & Restrictions or Word Rearrangements