Combinatorics · step 1 of 13

Sum & product rules

Two rules underpin all of counting. The sum rule: if a task can be done in one of several mutually exclusive ways, add the counts. The product rule: if a task is a sequence of independent choices, multiply. The word that decides between them is usually "or" versus "and" — but only when the alternatives really are disjoint, which is where inclusion–exclusion comes in.

Practise sum & product rules free → All Combinatorics subtopics

✓ Unlimited questions · marked criterion by criterion · no card needed

Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Decide: alternatives or stages?Mutually exclusive alternatives means add. A sequence of choices means multiply.
Check disjointness before addingThe sum rule requires no overlap. Overlapping cases need inclusion–exclusion instead.
Check independence before multiplyingThe number of options at each stage must not depend on which earlier choices were made — or must at least be constant.
Combine the rules for complex problemsSplit into disjoint cases (add), and count each case as a sequence of choices (multiply).

Worked example

A menu has 4 starters, 6 mains and 3 desserts. How many two-course meals consisting of a starter and a main, OR a main and a dessert?

Starter and main: a sequence of two choices, so 4 × 6 = 24 by the product rule.
Main and dessert: 6 × 3 = 18.
The two kinds of meal are mutually exclusive — one has a starter, the other a dessert.
So add: 24 + 18.

Answer. 42 two-course meals.

Where marks get dropped

These are the specific errors that cost credit on sum & product rules questions — QED's rubric penalises each of them separately.

Adding when the cases overlap. Any meal counted in both categories would be double-counted, so disjointness must be checked.
Multiplying when the number of options at the second stage depends on the first. Then split into cases first.
Reading "or" as always meaning add. Inclusive "or" over overlapping sets requires subtracting the intersection.

Practise this until it is automatic

Unlimited fresh questions

QED generates new sum & product rules problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.

Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.

Saved to your library

Every question you generate is kept and re-takeable as a timed exam, and your Combinatorics mastery is tracked so you know when this is exam-ready.

Sum & product rules — frequently asked questions

How do I know if cases are disjoint?

Ask whether a single object could be counted in two different cases. If yes, they overlap and the sum rule does not apply directly.

What if the second choice depends on the first?

If the NUMBER of options is constant, multiplication still works — arranging 5 people in a row gives 5 × 4 × 3 × 2 × 1 even though the specific options change.

How does this relate to sets?

The sum rule is |A ∪ B| = |A| + |B| for disjoint sets; the product rule is |A × B| = |A||B|. Counting is set theory with numbers.

The rest of Combinatorics

Counting principles, permutations, combinations. Each subtopic below has its own method, worked example and mark-losing traps.

Ready to make sum & product rules exam-proof?

Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.

Start practising free →