← Topic 3 · Counting & Probability
AMC 10/12 · Cheatsheet

Topic 3 · Counting & Probability

Chapter 1 · Counting techniques

📋 Reference · always available
Complementary
P(at least one)=1−P(none)P(\text{at least one}) = 1 - P(\text{none})
Stars & bars (≥0)
x1+⋯+xk=n⇒(n+k−1k−1)x_1+\cdots+x_k=n \Rightarrow \binom{n+k-1}{k-1}
Stars & bars (≥1)
(n−1k−1)\binom{n-1}{k-1}
Pre-assign one to each box first.
Inclusion–Exclusion
∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A\cup B|=|A|+|B|-|A\cap B|
Three-set PIE
+singles −pairs +triple. 'Divisible by' overlaps use the LCM.
Pigeonhole (basic)
nn pigeons in mm holes, n>mn>m ⇒ some hole has ≥2\ge 2.
Pigeonhole (general)
n pigeons in m holes  ⇒  ∃ hole with ⌈n/m⌉n\text{ pigeons in }m\text{ holes}\;\Rightarrow\;\exists\text{ hole with }\lceil n/m \rceil
Casework rule
Split on the tightest constraint. Cases must be exclusive AND exhaustive.
When to switch
More than 3 casework branches? Try complementary counting or bijection first.