Counting strategies (AQA GCSE Maths): Revision Notes
Counting strategies
What are counting strategies?
Counting strategies are systematic methods used to find all possible combinations or arrangements in a given situation. When faced with problems involving multiple choices or options, these strategies help ensure you don't miss any possibilities and can organise your work clearly.
The key to successful counting is being systematic - this means following an organised approach rather than randomly listing possibilities.
Being systematic is crucial in counting problems. Random listing often leads to missed combinations or duplicate counting, which can cost valuable marks in exams.
The systematic listing method
When you need to find all possible combinations, follow this structured approach:
Step 1: Identify the categories or positions you're working with Step 2: Fix one element in the first position Step 3: List all possibilities for the remaining positions Step 4: Move to the next element in the first position Step 5: Repeat until all first-position options are covered Step 6: Count your total combinations
Example: Three-digit numbers
If you have three number cards (4, 5, 6) and want to make different three-digit numbers, organise your work like this:
- Numbers starting with 4: 456, 465
- Numbers starting with 5: 546, 564
- Numbers starting with 6: 645, 654
This systematic approach ensures you find all 6 possible combinations without missing any or counting duplicates.
Worked Example: Badge Combinations
A badge maker offers three colours (Red, Green, Blue) and three shapes (Circle, Star, Oval). How many different badge combinations are possible?
Solution: Using systematic listing, organise by colour first:
- Red badges: RC, RS, RO
- Green badges: GC, GS, GO
- Blue badges: BC, BS, BO
Answer: 9 possible combinations
This method shows clearly that each colour can be paired with each shape, giving us total possibilities.
Worked Example: Sports Tournament
A hockey league has 5 teams. If each team plays every other team exactly once, how many matches will be played in total?
Solution: List systematically by team:
- Team 1 plays: Teams 2, 3, 4, 5 (4 matches)
- Team 2 plays: Teams 3, 4, 5 (3 remaining matches)
- Team 3 plays: Teams 4, 5 (2 remaining matches)
- Team 4 plays: Team 5 (1 remaining match)
Answer: matches total
Notice how we avoid double-counting by only listing forwards matches from each team.
Exam tips for counting strategies
Show your systematic approach
- Always demonstrate your method clearly in your working
- Use organised lists or tables to present your combinations
- Number your items to make counting easier
Even if you're confident in your counting, always show your systematic working. Examiners can award partial marks for correct method even if the final answer is wrong.
Save time and avoid errors
- You can number items in your list (1, 2, 3...) to speed up counting
- This still shows you're being systematic while being more efficient
- Read questions carefully to understand what's being asked
Common Mistakes to Avoid:
- Double-counting: Make sure each combination appears only once
- Missing combinations: Use a systematic order to ensure completeness
- Misreading the question: Check whether you need arrangements or combinations
These mistakes are easily avoided by following a clear systematic approach and checking your work carefully.
When to use counting strategies
Use systematic counting when problems involve:
- Choosing combinations from multiple categories
- Tournament schedules where teams play each other
- Menu combinations or outfit choices
- Any situation asking "how many different ways" or "how many combinations"
Key Points to Remember:
- Be systematic - follow an organised method rather than listing randomly
- Show your working - demonstrate your systematic approach in exams
- Check for completeness - ensure you haven't missed any possibilities
- Avoid double-counting - each combination should appear only once
- Read carefully - understand exactly what the question is asking for