AMC 10/12 · Topic Cheatsheet

Topic 3 · Counting & Probability

23 key results accumulated across 3 chapters.

Complementary

Ch 1

P(\text{at least one}) = 1 - P(\text{none})

Stars & bars (≥0)

Ch 1

x_1+\cdots+x_k=n \Rightarrow \binom{n+k-1}{k-1}

Stars & bars (≥1)

Ch 1

\binom{n-1}{k-1}

Pre-assign one to each box first.

Inclusion–Exclusion

Ch 1

|A\cup B|=|A|+|B|-|A\cap B|

Three-set PIE

Ch 1

+singles −pairs +triple. 'Divisible by' overlaps use the LCM.

Pigeonhole (basic)

Ch 1

n

pigeons in

m

holes,

n>m

⇒ some hole has

\ge 2

Pigeonhole (general)

Ch 1

n\text{ pigeons in }m\text{ holes}\;\Rightarrow\;\exists\text{ hole with }\lceil n/m \rceil

Casework rule

Ch 1

Split on the tightest constraint. Cases must be exclusive AND exhaustive.

When to switch

Ch 1

More than 3 casework branches? Try complementary counting or bijection first.

Product / sum rule

Ch 2

\text{AND}\Rightarrow\times,\;\; \text{OR}\Rightarrow +

Permutations

Ch 2

{}_nP_r = \tfrac{n!}{(n-r)!}

Combinations

Ch 2

\binom{n}{r} = \tfrac{n!}{r!(n-r)!} = \tfrac{{}_nP_r}{r!}

Basic

Ch 3

P = \tfrac{\#\text{favorable}}{\#\text{total}}

Independent

Ch 3

P(A\text{ and }B) = P(A)P(B)

Complement

Ch 3

P(\ge1) = 1 - P(\text{none})

Geometric

Ch 3

Probability = favorable area ÷ total area.

Conditional

Ch 3

P(A|B) = \tfrac{P(A\cap B)}{P(B)}

Bayes

Ch 3

P(A|B) = \tfrac{P(B|A)\,P(A)}{P(B)}

Total probability

Ch 3

P(B) = P(B|A)P(A) + P(B|\neg A)P(\neg A)

Markov recursion

Ch 3

p_k = \sum_\ell P(\text{transition }k\!\to\!\ell)\,p_\ell

with boundary conditions; solve linearly or telescope.

Weighted mean

Ch 3

\bar x = \tfrac{\sum w_i x_i}{\sum w_i}

Median

Ch 3

2k{+}1

values: middle =

(k{+}1)^{\text{th}}

2k

values: avg of

k^{\text{th}}, (k{+}1)^{\text{th}}

Unique mode

Ch 3

Strict inequality: every other value appears strictly fewer times.