Systematic Listing
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Revision notes for WJEC GCSE Maths Systematic Listing. Open the guide for explanations and worked examples. Written against the WJEC GCSE Maths (3320QD) specification, so the content matches what's examinable rather than general Maths background.

Systematic Listing

What you'll learn

  • How to write organised lists so you do not miss choices.
  • How to handle menus, coins, dice and number cards.
  • How to decide when order changes the answer.
  • How to use a full list to answer chance questions.

1. The basic idea

Systematic listing means making your list in a clear pattern. The pattern is the important part: it helps you avoid missing answers or writing the same answer twice.

Definition

Key words

  • An outcome is one possible result, such as getting heads when you toss a coin.
  • A combination is a choice made from two or more groups, such as one sandwich and one drink.
  • The sample space is the complete list of all possible outcomes.
  • Systematic listing means writing the sample space in an organised order.
Key Idea

The main rule

Keep one choice fixed, list everything that goes with it, then move to the next choice.

Example

Choosing one item from each list

A cafe offers two sandwiches: Cheese or Tuna. It also offers three drinks: Tea, Juice or Water. List all possible lunch choices.

A branching diagram shows each sandwich paired with every drink to make 6 lunch combinations.

  1. Start with Cheese and pair it with every drink: Cheese and Tea, Cheese and Juice, Cheese and Water.

  2. Now move to Tuna and pair it with every drink: Tuna and Tea, Tuna and Juice, Tuna and Water.

  3. Count the list. There are 6 combinations, which makes sense because there are 2 sandwich choices and 3 drink choices.

2. Lists with more than one action

Some questions involve doing two actions, such as rolling a dice and flipping a coin. Your outcome must include both parts.

A good way is to fix the first result, then list all the second results.

Tip

Use short labels

For a coin, you can write H for heads and T for tails. This keeps your list neat.

Example

A number spinner and a coin

A spinner can land on 1, 2, 3 or 4. A coin can land on H or T. List all possible outcomes.

A two-column systematic list keeps each spinner result fixed while pairing it with H and T.

  1. Start with spinner result 1: 1H, 1T.

  2. Move to spinner result 2: 2H, 2T.

  3. Move to spinner result 3: 3H, 3T.

  4. Move to spinner result 4: 4H, 4T.

  5. The full list is: 1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T.

3. Repeating the same action

If you toss a coin three times, each toss has to be recorded. The order is important.

Definition

Order matters

Order matters when changing the order gives a different result. For example, HHT is different from HTH because the tails happens on a different toss.

Common Mistake

Stopping too early

For three coin tosses, do not just write HHH and TTT. You need every possible mix of H and T in every position.

Example

Tossing a coin three times

A coin is tossed three times. List all possible outcomes.

A three-stage tree diagram records every possible sequence of heads and tails.

  1. First list the outcomes for two tosses: HH, HT, TH, TT.

  2. Add a third toss to HH: HHH, HHT.

  3. Add a third toss to HT: HTH, HTT.

  4. Add a third toss to TH: THH, THT.

  5. Add a third toss to TT: TTH, TTT.

  6. The full sample space has 8 outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

4. Number cards

When you make numbers from cards, each card can usually be used once, unless the question says otherwise.

Definition

Digit

A digit is a single symbol used to make numbers, such as 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9.

Common Mistake

Using a card twice

If the cards are 4, 6 and 9, the number 44 is not allowed because there is only one 4 card.

Example

Making two-digit numbers

The cards show 4, 6 and 9. Write down all the two-digit numbers you can make.

A place-value layout shows each card used once to form the allowed two-digit numbers.

  1. Put 4 in the tens position. The units digit can be 6 or 9: 46, 49.

  2. Put 6 in the tens position. The units digit can be 4 or 9: 64, 69.

  3. Put 9 in the tens position. The units digit can be 4 or 6: 94, 96.

  4. The full list is: 46, 49, 64, 69, 94, 96.

Repeated digits

Sometimes two cards show the same digit. The cards are separate, but if the final numbers look the same, only write the number once when the question asks for different possible numbers.

Example

Making numbers with repeated digits

The cards show 2, 2 and 5. Write all the different three-digit numbers you can make.

Separate cards with repeated digits can give duplicate arrangements, so only the different final numbers are listed.

  1. Start with 2. The other two digits are 2 and 5, so you can make 225 and 252.

  2. Start with 5. The remaining digits are 2 and 2, so you can only make 522.

  3. The different numbers are: 225, 252, 522.

5. When order does not matter

In some lists, swapping the order does not create a new answer. This often happens with matches between teams.

For example, Lions v Tigers is the same match as Tigers v Lions, so only write it once.

Common Mistake

Double-counting matches

In a match list, A v B and B v A are the same match unless the question says the teams play twice.

Example

Listing matches

Four clubs are called Lions, Tigers, Eagles and Sharks. Each club plays every other club once. List the matches.

A match diagram connects each pair of clubs once, avoiding repeated fixtures such as Tigers v Lions.

  1. Start with Lions: Lions v Tigers, Lions v Eagles, Lions v Sharks.

  2. Move to Tigers. Do not repeat Tigers v Lions, because it is already listed. Write Tigers v Eagles, Tigers v Sharks.

  3. Move to Eagles. Do not repeat Eagles v Lions or Eagles v Tigers. Write Eagles v Sharks.

  4. The matches are: Lions v Tigers, Lions v Eagles, Lions v Sharks, Tigers v Eagles, Tigers v Sharks, Eagles v Sharks.

6. Using a list for chance questions

Once you have a complete list, you can count how many outcomes match the question.

Definition

Chance words

  • Equally likely means each outcome has the same chance of happening.
  • A favourable outcome is an outcome that matches what the question wants.
  • Probability is the chance of something happening. For equally likely outcomes, probability is number of favourable outcomestotal number of outcomes\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}total number of outcomesnumber of favourable outcomes​.
Example

Cards from two boxes

Box A has cards 1, 4 and 7. Box B has cards 3, 5 and 6. One card is taken from each box and the numbers are multiplied. Find the probability that the answer is odd.

A 3 by 3 sample-space grid highlights the pairs where both cards are odd.

  1. List the possible pairs: 1 with 3, 1 with 5, 1 with 6; 4 with 3, 4 with 5, 4 with 6; 7 with 3, 7 with 5, 7 with 6.

  2. An odd product happens when both numbers are odd.

  3. The favourable pairs are: 1 with 3, 1 with 5, 7 with 3, 7 with 5. There are 4 favourable outcomes.

  4. There are 9 possible pairs altogether.

  5. The probability is 49\frac{4}{9}94​.

Tip

For two dice

A grid is often neater than a long list for two dice. Write the first dice down the side and the second dice across the top, then count the results you need.

A dice grid is a compact way to organise all 36 outcomes for two dice.

Exam technique

In the exam

  1. Read the wording carefully and decide whether order matters.

  2. Keep one choice fixed and work through all the other choices before moving on.

  3. Count your list at the end. For probability questions, use favourable outcomes over total outcomes.

Self review

Check yourself

  • If there are 3 starters and 2 mains, how many meal combinations should your list contain?
  • Why is HHT different from HTH when a coin is tossed three times?
  • With cards 1, 1 and 4, what different three-digit numbers can you make?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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