Ever wondered how many different outfits you can make, or how many unique phone numbers are possible? These questions are about counting possibilities, and in maths, we use a powerful tool called the Product Rule for Counting to find the answer without listing every single option.
What is the Product Rule?
If there are $m$ ways to do one task and $n$ ways to do a second, independent task, then there are a total of $m \times n$ ways to do both tasks.
For example, if you have 3 different shirts and 2 different pairs of trousers, the total number of outfits is simply $3 \times 2 = 6$. A tree diagram is a great way to visualise this:
The rule extends for more tasks. If you also had 4 pairs of shoes, the total combinations would be $3 \times 2 \times 4 = 24$.
Worked Examples
Example 1: Meal Deals
A café offers 5 sandwiches, 4 drinks, and 3 snacks. How many different meal deal combinations are possible?
- Identify the choices for each independent task.
- Apply the product rule: choices are made for a sandwich AND a drink AND a snack.
Calculation:
$5 \times 4 \times 3 = 60$
Answer: 60 combinations
Example 2: PIN Code
How many different 4-digit PINs are possible using digits 0–9, if digits can be repeated?
- There are 4 positions to fill.
- For each position, there are 10 possible digits (0 through 9).
Calculation:
$10 \times 10 \times 10 \times 10 = 10^4$
Answer: 10,000 PINs
Example 3: No Repetition
How many 3-letter arrangements can be made from A, B, C, D, E if each letter is used only once?
- 1st letter: 5 choices.
- 2nd letter: One letter used, so 4 choices remain.
- 3rd letter: Two letters used, so 3 choices remain.
Calculation:
$5 \times 4 \times 3 = 60$
Answer: 60 arrangements
Tutor Insights
For your GCSE exams, you need to recognise when to use the product rule, especially in word problems. Pay close attention to whether repetition is allowed, as this is a key detail.
🤔 Common Misunderstandings
- Adding instead of multiplying: Students often add the options. Remember, if you are making one choice AND another, you multiply.
- Forgetting about repetition: Always check if items can be used more than once. If not, the number of options will decrease with each choice.
📝 Common Exam Mistakes
- Simple calculation errors: Double-check your multiplication.
- Misinterpreting the question: Not reading carefully if there are special conditions, like “the code must start with an odd number”.
- Not showing your working: Write down the multiplication sum (e.g., $5 \times 4 \times 3$) to get method marks.
Practice Questions
- Easy A café offers 6 types of coffee and 4 types of cake. How many ways can a customer choose one coffee and one cake?
- Medium A safe requires a 3-digit code using digits from 0–9.
a) How many codes are possible if digits can be repeated?
b) How many are possible if digits cannot be repeated? - Harder How many even 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5 without repetition? (Hint: Consider the choice for the last digit first!)
Show Answers
- Working: $6 \times 4 = 24$.
Answer: 24 ways. -
a) With repetition: $10 \times 10 \times 10 = 1{,}000$.
Answer: 1,000 codes.
b) Without repetition: $10 \times 9 \times 8 = 720$.
Answer: 720 codes. - Working: To be even, the last digit must be 2 or 4 (2 choices). Once chosen, there are 4 digits left for the first position, then 3 for the second. Total $= 2 \times 4 \times 3 = 24$.
Answer: 24 numbers.
FAQs
Q: When do I add instead of multiply?
A: You multiply when combining choices (e.g., a shirt AND trousers). You add when choosing between alternatives (e.g., choosing a cake OR a biscuit). The keyword “AND” usually means multiply.
Q: Can I use the product rule for more than two things?
A: Absolutely! You just keep multiplying the number of options for each task. If you have choices for Task 1, Task 2, Task 3, and so on, you multiply all the options together.
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