Advanced Counting & Probability

1. Introduction

Counting problems are everywhere on the AMC 8: how many ways can a class line up, how many committees can be formed, what is the chance two dice sum to $7$? At first these feel like a grab-bag of tricks, but almost all of them flow from a single idea — the multiplication principle — plus two refinements, permutations (order matters) and combinations (order doesn't). Probability then sits right on top of counting: it is just favorable outcomes divided by total outcomes.

The skill that separates strong contestants is knowing which tool to reach for. Should you multiply? Should you divide out repeats? Should you count the opposite event instead? A surprising number of "hard" problems become one-line computations once you pick the right framing — especially using complementary counting, where you count what you don't want and subtract.

This article develops the full toolkit: the multiplication and addition principles, permutations and combinations (with the factorial machinery behind them), arrangements with repeated items, complementary counting, casework, expected value basics, and the basic laws of probability for independent and mutually exclusive events. Each technique comes with contest-style examples so you can see exactly when it applies.

2. Core Concepts

Concept 1 — The Multiplication Principle

Multiplication principle. If a task is done in stages, and stage one can happen in $a$ ways and stage two in $b$ ways (independent of the first), then the whole task happens in $a\cdot b$ ways. This extends to any number of stages. The keyword "and" (in the sense of sequential choices) usually signals multiplication.

Concept 2 — The Addition Principle

Addition principle. If a task can be done by one of several mutually exclusive methods — $a$ ways by method one or $b$ ways by method two, with no overlap — then there are $a + b$ ways total. The keyword "or" (exclusive) usually signals addition.

Concept 3 — Factorials

A factorial $n! = n\cdot(n-1)\cdots 2\cdot 1$ counts the arrangements of $n$ distinct objects in a row. By convention $0! = 1$, which keeps all combination formulas consistent. Know $5! = 120$, $6! = 720$, $7! = 5040$.

Concept 4 — Permutations

A permutation counts ordered selections. The number of ways to arrange $r$ of $n$ distinct items in order is $P(n,r) = \frac{n!}{(n-r)!} = n(n-1)\cdots(n-r+1).$ The full permutation of all $n$ items is $n! = P(n,n)$.

Concept 5 — Combinations

A combination counts unordered selections. The number of ways to choose $r$ of $n$ items, ignoring order, is $\binom{n}{r} = \frac{n!}{r!\,(n-r)!} = \frac{P(n,r)}{r!}.$ The extra division by $r!$ removes the orderings that a combination regards as identical.

Concept 6 — Pascal's Rule and Symmetry

Symmetry: $\binom{n}{r} = \binom{n}{n-r}$. Choosing what to include is the same as choosing what to leave out.

Pascal's rule: $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$, the recurrence behind Pascal's triangle. Useful for quick mental arithmetic: $\binom{8}{3} = \binom{7}{2} + \binom{7}{3} = 21 + 35 = 56$.

Concept 7 — Arrangements with Repeated Objects

When some objects are identical, divide out the permutations of the identical copies: $\frac{n!}{a!\,b!\cdots}$ where $n$ is the total count and $a, b, \ldots$ are the multiplicities of each repeated item.

Concept 8 — Probability Fundamentals

The probability of an event $E$ in a finite sample space $S$ of equally likely outcomes is $P(E) = \frac{|E|}{|S|} = \frac{\text{favorable outcomes}}{\text{total outcomes}}.$ Probabilities lie between $0$ and $1$. The complement $E^c$ is "$E$ does not happen," and $P(E) = 1 - P(E^c)$.

Concept 9 — Independent Events

Two events are independent if one happening doesn't change the other's probability; then $P(A \cap B) = P(A)\cdot P(B)$. Drawing with replacement or rolling separate dice are classic independent setups.

Concept 10 — Mutually Exclusive Events

Two events are mutually exclusive if they cannot both happen; then $P(A \cup B) = P(A) + P(B)$. For events that can overlap, use inclusion–exclusion: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Concept 11 — Complementary Counting

When a problem says "at least one," count the complement ("none") and subtract from the total. This is almost always faster than casework on "exactly one, exactly two, …"

Concept 12 — Expected Value (AMC 8 Preview)

The expected value of a random variable $X$ taking values $x_1, x_2, \ldots$ with probabilities $p_1, p_2, \ldots$ is $E[X] = \sum x_i p_i.$ Linearity of expectation: $E[X + Y] = E[X] + E[Y]$ even when $X$ and $Y$ are not independent. This solves many "average number of …" problems without full casework.