X Tables
Problem Solving

Systematic listing

List possibilities without missing or repeating any.

Quick answer

Systematic listing means writing down all the possible answers to a problem in a careful, ordered way so you don’t miss any or write the same one twice.

Learning outcome

By the end of this page, you will be able to list all the possible answers to a problem in order, so you know you haven’t missed anything or repeated anything. Success signs:
  • You can think of all the different ways to solve a problem
  • You can write them down in a clear order
  • You can check that you haven’t missed or repeated any answers
  • You can use your list to help you find the right answer

Why this matters

Systematic listing is a helpful way to solve problems when you have more than one answer. It helps you think carefully and not make mistakes. This skill is useful in maths, puzzles, games, and even in everyday life. For example, if you’re planning a party and need to choose 2 snacks from 4 options, you need to list all the combinations without missing any or repeating the same pair.

How it works

To list things systematically:
  1. First, understand what you are looking for
  2. Make a plan to list them in a clear order
  3. Go through each possibility one by one
  4. Check your list to make sure you haven’t missed or repeated anything

Examples you can see

Let’s imagine you have 3 toy cars and 2 toy boats. You want to know how many different ways you can pick one car and one boat. Step 1: List the cars: Car A, Car B, Car C Step 2: List the boats: Boat 1, Boat 2 Step 3: Now match each car with each boat:
  • Car A + Boat 1
  • Car A + Boat 2
  • Car B + Boat 1
  • Car B + Boat 2
  • Car C + Boat 1
  • Car C + Boat 2
That’s 6 different combinations. You can see that we listed each car with each boat, so we didn’t miss any and didn’t repeat any.

Try it as a mini adventure

Adventure: The Treasure Hunt You are on a treasure hunt with your friend. You need to collect 3 different items from a list of 5: a red key, a blue coin, a green leaf, a yellow rock, and a purple shell. Challenge: Find all the different ways you can pick 3 items from the 5. How to start:
  • Write down all 5 items
  • Choose 3 items at a time
  • Make sure you list each group once only
Try this:
  1. Start with the red key: red key + blue coin + green leaf
  2. Next, red key + blue coin + yellow rock
  3. Next, red key + blue coin + purple shell
  4. Next, red key + green leaf + yellow rock
  5. Next, red key + green leaf + purple shell
  6. Next, red key + yellow rock + purple shell
  7. Now, blue coin + green leaf + yellow rock
  8. Next, blue coin + green leaf + purple shell
  9. Next, blue coin + yellow rock + purple shell
  10. Finally, green leaf + yellow rock + purple shell
You have 10 different combinations. That’s a lot of possibilities, but by listing them carefully, you didn’t miss any.

Common mistakes

  • Mistake: Listing the same combination twice Why it happens: Not checking your list carefully Fix: Compare each new item with the ones you already have
  • Mistake: Forgetting to check all the possibilities Why it happens: Jumping to conclusions too quickly Fix: Use a system like starting with one item and pairing it with all others
  • Mistake: Mixing up the order of items Why it happens: Not writing in a clear, ordered way Fix: Always list items in the same order (e.g., alphabetical or by number)

Parent and carer guide

Parents and carers can help by:
  • Encouraging children to use a system, like starting with one item and listing all the others with it
  • Checking that the list is ordered and complete
  • Practising with simple examples like choosing 2 toys from 4
  • Letting children make mistakes and correct them themselves
  • Using everyday situations like planning a meal or choosing clothes to practice listing
What to say:
  • “Let’s think of all the ways to do this step by step.”
  • “Can you show me how you listed your choices?”
  • “Do you think you’ve got them all?”
What not to worry about:
  • Children may make small errors at first, but they learn quickly
  • It’s okay to go back and correct mistakes
  • Don’t rush – take your time to make sure you get it right

Practice questions

  1. How many different ways can you pick 2 letters from the letters A, B, C?
  2. List all the possible 2-digit numbers you can make using the digits 1, 2, 3 (you can use each digit only once).
  3. If you have 4 different coloured pencils (red, blue, green, yellow), how many ways can you pick 2 pencils?
  4. How many different ways can you choose 3 toys from 5 toys (A, B, C, D, E)?
  5. List all the possible 3-digit numbers you can make using the digits 1, 2, 3 (you can use each digit only once).
Answers:
  1. 3 ways: AB, AC, BC
  2. 12, 13, 21, 23, 31, 32
  3. 6 ways: red-blue, red-green, red-yellow, blue-green, blue-yellow, green-yellow
  4. 10 ways: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE
  5. 123, 132, 213, 231, 312, 321

FAQ

What is systematic listing?

Systematic listing is a way of writing down all the possible answers to a problem in a clear, ordered way so you don’t miss any or repeat any.

Why do we need to list things systematically?

It helps us be sure we have found all the answers to a problem. It’s especially helpful when there are many possible answers.

Can you give me a real-life example?

Yes! If you are choosing 2 snacks from 4 options (apple, banana, cookie, cake), you can list them systematically to make sure you don’t miss any combination.

How can I practice systematic listing?

You can practice by picking 2 or 3 items from a small group and listing all the possible ways to choose them. Try it with toys, clothes, or food.

What if I still make mistakes?

That’s normal! The important thing is to check your list and fix any mistakes. Practice makes progress.