If you can choose a shirt in 4 ways, trousers in 3 ways and shoes in 2 ways, there are 4 × 3 × 2 = 24 outfits. The same principle counts licence plates, passwords, menu combinations and lottery tickets.
How to use the Fundamental Counting Principle Calculator
- Number of choices to make
- Options for choice 1
- Options for choice 2
- Options for choice 3
- Options for choice 4
- Options for choice 5
- Options for choice 6
- Options for choice 7
- Options for choice 8
- Options for choice 9
- Options for choice 10
Results update as you type, and everything is calculated in your browser.
How it works
Each independent choice multiplies the number of possibilities. Enter how many options there are at each step.
Inputs explained
| Field | What to enter |
|---|---|
| Number of choices to make | Choose one: 2, 3, 4, 5 … |
| Options for choice 1 | A number |
| Options for choice 2 | A number |
| Options for choice 3 | A number |
| Options for choice 4 | A number |
| Options for choice 5 | A number |
| Options for choice 6 | A number |
| Options for choice 7 | A number |
| Options for choice 8 | A number |
| Options for choice 9 | A number |
| Options for choice 10 | A number |
Example values
How the main result changes when Options for choice 1 changes and everything else stays at the example values above:
| Options for choice 1 | Total number of outcomes |
|---|---|
| 2 | 12 |
| 3 | 18 |
| 4 | 24 |
| 5 | 30 |
| 6 | 36 |
| 8 | 48 |
Formula
Total outcomes = n₁ × n₂ × … × nₖ
Worked example
Inputs — Number of choices to make: 3; Options for choice 1: 4; Options for choice 2: 3; Options for choice 3: 2.
Results — Total number of outcomes = 24.
Key terms
- Outcome
- One possible result of a sequence of choices.
Frequently asked questions
What is the fundamental counting principle?
If one event can happen in m ways and another in n ways, together they can happen in m × n ways.
How many 4-digit PINs are there?
10 × 10 × 10 × 10 = 10,000.
Does order matter?
The principle counts ordered sequences of choices; use combinations when order does not matter.
What if choices depend on each other?
Use the number of options remaining at each step, e.g. 10 × 9 × 8 without repetition.