📏 Essential Formulas

Complete formula bank for AMC counting and probability problems with usage notes and micro-examples.

📚 Formula Categories

🔢 Basic Counting

• Addition Principle: $|A \cup B| = |A| + |B| - |A \cap B|$
• Multiplication Principle: $|A \times B| = |A| \cdot |B|$
• Complement: $|A^c| = |S| - |A|$

🔄 Permutations & Combinations

• Permutations: $P(n,k) = \frac{n!}{(n-k)!}$
• Combinations: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
• Circular permutations: $(n-1)!$ (no reflection) or $\frac{(n-1)!}{2}$ (with reflection)
• Multinomial: $\binom{n}{k_1,k_2,\ldots,k_r} = \frac{n!}{k_1!k_2!\cdots k_r!}$

🎯 Advanced Counting

• Stars and bars (nonnegative): $\binom{n+k-1}{k-1}$
• Stars and bars (positive): $\binom{n-1}{k-1}$
• Inclusion-exclusion (2 sets): $|A \cup B| = |A| + |B| - |A \cap B|$
• Inclusion-exclusion (3 sets): $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$
• Pigeonhole bound: At least $\lceil \frac{n}{k} \rceil$ objects in one box

🎲 Probability

• Basic probability: $P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$
• Conditional probability: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$
• Independence: $P(A \cap B) = P(A)P(B)$
• Law of total probability: $P(A) = \sum_i P(A \mid B_i)P(B_i)$
• Bayes’ theorem: $P(B_i \mid A) = \frac{P(A \mid B_i)P(B_i)}{P(A)}$

📊 Expected Value

• Expected value: $\mathbb{E}[X] = \sum_x x \cdot P(X = x)$
• Linearity: $\mathbb{E}[X + Y] = \mathbb{E}[X] + \mathbb{E}[Y]$
• Indicators: $\mathbb{E}[I_A] = P(A)$

📈 Distributions

• Binomial: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$, $\mathbb{E}[X] = np$
• Geometric: $P(X = k) = (1-p)^{k-1} p$, $\mathbb{E}[X] = \frac{1}{p}$
• Hypergeometric: $P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$

🔄 Special Arrangements

• Derangements: $!n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}$

🎯 How to Use This Section

1. Quick Reference: Use as a lookup table during practice
2. Contest Prep: Memorize the most common formulas
3. Problem Solving: Identify which formula applies to your problem
4. Verification: Check your work against these formulas

📊 Formula Priority by Level

AMC 10 Essential

• Basic counting principles
• Permutations and combinations
• Basic probability
• Simple expected value

AMC 12 Essential

• Everything from AMC 10
• Advanced counting (stars and bars, inclusion-exclusion)
• Expected value and linearity
• Probability distributions

Next: Essential Formulas → Problem-Solving Tips

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