Fundamental counting principle

Definition

If one action can be done in n₁ ways and a second in n₂ ways, the two together can be done in n₁ × n₂ ways. Arrangements of n distinct objects total n!.

Worked example

A 4-digit PIN using digits 0–9 with repetition allowed has 10⁴ = 10 000 possibilities.

The letters of MATHS can be arranged in 5! = 120 ways. If M and A must stay together, treat MA as one unit: 4! × 2! = 48 arrangements.

Where it appears in the exam

Paper 1 counting questions ask for the number of codes or arrangements, often with conditions, and sometimes use the result to find a probability.

Step-by-step method

1. Count the positions or choices to fill.
2. Find the number of options for each position.
3. Multiply the options together.
4. Use n! to arrange n distinct objects, and treat items that must stay together as one unit.

Common mistake

Adding the options instead of multiplying them.

Related probability terms

• Mutually exclusive events
• Independent events

Frequently asked questions

What does fundamental counting principle mean in Grade 12 Maths?

If one action can be done in n₁ ways and a second in n₂ ways, the two together can be done in n₁ × n₂ ways. Arrangements of n distinct objects total n!.

Which NSC paper tests fundamental counting principle?

Fundamental counting principle belongs to the Probability section of CAPS Grade 12 Mathematics, which is examined in Paper 1.

How do I practise questions involving fundamental counting principle?

Snap a question that uses fundamental counting principle into the Snap&Learn AI solver, or work through the probability topic guide and NSC past papers.