Counting Principles

Rule of product (and) and rule of sum (or).

The idea

The rule of product and the rule of sum are the two principles from which counting arguments are built. We break a count into smaller choices, then multiply or add, according to whether the choices are made one after another or instead of one another.

Proposition (Rule of product).

If a task consists of a first choice with $m$ options followed by a second choice with $n$ options, where the number of options at the second stage does not depend on the first choice, then the task can be carried out in $m \times n$ ways.

Proposition (Rule of sum).

If a task consists of a single choice from one of two disjoint groups, one with $m$ options and the other with $n$, then the task can be carried out in $m + n$ ways.

To illustrate the rule of product, suppose a task consists of one choice and then a second: a shirt from $m$ shirts, then trousers from $n$ pairs. Every shirt goes with every pair of trousers, so the outfits number $m \times n$. The rule extends to any number of choices made in sequence, provided the number of options at each stage does not depend on the earlier choices.

To illustrate the rule of sum, suppose instead the task is a single choice from one of two groups: a drink that is one of $m$ teas or one of $n$ coffees. The possibilities number $m + n$. The groups must be disjoint, since an item in both would be counted twice.

In a problem statement, "and" or "then" calls for multiplying; "or", meaning exactly one of the cases, calls for adding.

Ways to work on it

• Walkthrough. Product vs sum: when choices multiply and when they add.
• Practice. Apply the product or sum rule.
• Hardest. Count the arrangements that contain a repeated digit.

Not sure where to start? Take the ten-question placement test.