Counting¶

Definition¶

Let $A$ be a finite set. Counting is the process of determining the number of elements of $A$ , usually by using systematic rules such as addition, multiplication, permutations, or combinations.

Interpretation¶

Counting methods help determine how many outcomes are possible. They are essential when probabilities are based on equally likely outcomes.

A tree with two shirts on the first level and three trousers below each, giving six outfits at the leaves.

The diagram pairs each of two shirts with each of three pairs of trousers, giving $2\cdot3=6$ outfits. Multiplying the numbers of choices counts these ordered combinations; whether order matters determines when permutations or combinations are appropriate.

Finite and infinite sets distinguishes a finite outcome count from an unbounded or countably infinite collection. Cardinality is the number being computed; the counting rules provide ways to find it without listing every element.

Key Results¶

The counting multiplication rule multiplies the number of choices at successive stages: if every first-stage choice has $m$ second-stage choices and there are $n$ first-stage choices, there are $nm$ ordered outcomes.
The addition rule counts alternatives that do not overlap: for disjoint finite sets, $|A\cup B|=|A|+|B|$ $∣ B ∣$ . Overlapping alternatives need the inclusion–exclusion principle.
Counting arguments support naive probability models when outcomes are equally likely.

Examples¶

A password with two letters followed by three digits has $26^2\cdot10^3=676{,}000$ possibilities when all 26 letters and 10 digits are allowed at each position and repetition is permitted.
If a menu offers either a soup or a salad but not both, the addition rule counts the first-course choices.
Counting favourable poker hands and total poker hands gives a naive probability.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 1

Keywords: combinatorics, enumerative combinatorics Symbols: union

Last updated: 20 October 2026 03:02