Example 1
In how many ways can three friends A, B and C stand in a line?
- Start with A first: ABC, ACB.
- Then B first: BAC, BCA.
- Then C first: CAB, CBA.
- That is 3 groups of 2.
Answer: 6 ways
🏅 Olympiad maths · session 3 of 49 · 60 minutes · ⭐ Plus
Counting without listing: the multiplication principle
How many outfits, codes or numbers can you make? Learn to count them without writing every one: list in order, draw a tree, then multiply the choices.
A complete lesson · 60 minutes
Teach it, see it, practise it, all on this page.
Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.
- Warm-up4 min
- Organised lists8 min
- Trees9 min
- Multiply7 min
- Rules7 min
- Activity 17 min
- Activity 27 min
- Activity 36 min
- Quiz4 min
- Set the project1 min
By the end you can
- Make an organised list so nothing is missed or repeated
- Draw a tree diagram for choices made one after another
- Use the multiplication principle
- Count when there are rules (no repeats, must be even, no zero first)
Before you start
- Multiplication facts
- Place value: hundreds, tens and ones
Have ready
- Paper and pencil
- Three coloured pencils
Classes 4 to 7 · ages 9 to 13
Warm-up · think first
You have 2 T-shirts (red, blue) and 3 pairs of shorts (black, grey, white). How many different outfits can you wear? Try to list them all.
Show the answer
6 outfits: red with each of the 3 shorts, and blue with each of the 3 shorts. 2 × 3 = 6. Today you will learn why multiplying works, and how it saves time when there are hundreds of outfits.
Teacher tip: Let them list freely first. Then ask: how can we be sure we did not miss one? That leads to listing in order.
Part 1 of 4 · learn, see, practise
Make an organised list
When you list possibilities, fix the first choice and go through every second choice, then move to the next first choice. This stops you missing or repeating any.
Arrangements of things in a line are a good example: write them in alphabetical order.
Think first: List all the 2-digit numbers you can make from the digits 1, 2 and 5 without repeating a digit.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Warm-up
Core
Stretch 🔥
Answers (for the teacher)
- 6
- 6
- 2
- 12
⭐ Part of Pure Faith Plus
The rest of this lesson is in Plus
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You have seen the plan, the warm-up and part 1. With Plus you also get:
- Part 2: Tree diagrams, with practice at three levels
- Part 3: Multiply the choices, with practice at three levels
- Part 4: Counting with rules, with practice at three levels
- Class activity 1: How many ways?
- Class activity 2: Paths on a grid
- Class activity 3: Design a number plate
- A 8-question quiz
- After-class project: Design a menu
- Homework and a printable practice sheet with answers
- Teach mode and the minute-by-minute teacher's plan
The first 2 lessons of every course, every game and all maths practice stay free.
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FAQ
Counting without listing: the multiplication principle: questions
What is the multiplication principle in counting?
If one choice can be made in a ways and the next in b ways, the two together can be made in a × b ways. It extends to any number of steps.
How many 3-digit numbers have all different digits?
9 × 9 × 8 = 648: 9 choices for the first digit (not 0), then 9, then 8.
When should I use a tree diagram?
When choices happen one after another and you want to see every possibility. It also shows why the multiplication principle works.
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