The product rule for counting says that if one choice can be made in m ways and a second choice in n ways, the two together can be made in m × n ways. It extends to any number of stages: multiply the number of options at each stage to find the total number of possible outcomes.
When do you use the product rule?
Use it whenever a question asks how many possible outcomes, arrangements, codes, routes or combinations there are, and the outcome is built up in stages. The giveaway phrases are "how many different…", "how many possible…" and "in how many ways…".
The rule replaces listing. A sample space diagram or a list works fine for eight outcomes, but not for eight thousand — and GCSE questions are deliberately sized so that listing is impractical.
How do you apply the product rule step by step?
- Break the outcome into stages. Ask: what decisions are made, and in what order? Each decision is one stage.
- Count the options at each stage. Write the number above or beside each stage so nothing is lost.
- Check whether earlier choices change later ones. If items can be reused, the count stays the same each time. If they cannot, it falls by one each stage.
- Multiply all the stage counts together.
- Sense-check the size of the answer. More stages or more options should always give a bigger number.
Worked example: a three-course menu
Question: A restaurant offers 4 starters, 6 main courses and 3 desserts. How many different three-course meals are possible?
There are three stages, and the choices are independent — picking a starter does not remove any mains.
4 × 6 × 3 = 72 different meals
Worked example: a four-digit PIN
Question: A bank card PIN is four digits, each from 0 to 9. How many different PINs are possible?
Each of the four stages has 10 options, and digits can repeat (1233 is a valid PIN).
10 × 10 × 10 × 10 = 10⁴ = 10,000 PINs
Now with no repeats: if all four digits must be different, each stage uses up one digit, so the options fall by one each time.
10 × 9 × 8 × 7 = 5,040 PINs
Reading which of these two the question wants is the single most important skill in this topic. Look for the words "different", "distinct", "cannot be repeated" or "each used once".
Worked example: a number plate
Question: A code consists of two letters followed by three digits. Letters and digits may be repeated. How many codes are possible?
There are 26 letters and 10 digits.
26 × 26 × 10 × 10 × 10 = 676,000 codes
Writing the stages out as a row of boxes before multiplying keeps the count honest:
| Stage | 1st letter | 2nd letter | 1st digit | 2nd digit | 3rd digit |
|---|---|---|---|---|---|
| Options | 26 | 26 | 10 | 10 | 10 |
How do you handle restrictions?
Restricted questions are worth the most marks, and there is one reliable rule: deal with the restricted stage first, even if it is not first in the outcome.
Question: A four-digit code is made from the digits 0–9, and repeats are allowed, but the first digit cannot be 0. How many codes are possible?
The first digit has only 9 options (1–9). The other three still have 10 each.
9 × 10 × 10 × 10 = 9,000 codes
Question: Three-digit numbers are formed using the digits 1, 2, 3, 4 and 5, with no digit repeated. How many of them are even?
To be even, the number must end in 2 or 4 — so start with the last digit.
- Last digit: 2 options (2 or 4)
- First digit: 4 options remain
- Middle digit: 3 options remain
2 × 4 × 3 = 24 even numbers
If you had started at the left, you could not have counted the final stage, because whether 2 and 4 were still available would depend on what you had already used. Restriction first, always.
How does this connect to probability?
Once you can count outcomes, you can find probabilities of the form:
$$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$$
From the example above, the probability that a randomly formed three-digit number (from 1–5, no repeats) is even is 24 ÷ 60 = 2/5, because the total number of such numbers is 5 × 4 × 3 = 60. Counting the numerator and the denominator with the same method keeps the two consistent.
Common mistakes to avoid
- Adding instead of multiplying. 4 + 6 + 3 = 13 answers a different question ("how many dishes are on the menu?"). Stages that happen together multiply.
- Keeping the count constant when items cannot repeat. If the four PIN digits must differ, the third stage has 8 options, not 10.
- Ignoring a restriction until the end. Handle the restricted stage first.
- Counting a stage twice. Two letters followed by three digits is five stages, not six.
- Forgetting that 0 is a digit. There are ten digits, 0 to 9, not nine.
Frequently asked questions
Is the product rule for counting on foundation tier as well as higher?
Yes. Counting outcomes by multiplying the options at each stage appears on both foundation and higher tier papers, usually within the probability and statistics section. Higher-tier versions tend to add a restriction, ask you to work with the answer as a probability, or combine the count with a "no repeats" condition, but the underlying rule is identical.
How do I know whether repeats are allowed?
Read the context and the exact wording. Real-world items that stay available allow repeats — a PIN digit, a letter on a number plate, a coin toss, a menu choice for different people. Items that are physically used up do not — dealing cards from a pack, picking pupils for different roles, or arranging books on a shelf. If the question says "different", "distinct" or "each used only once", repeats are not allowed.
Do I need to know factorials or nCr for this at GCSE?
No. The GCSE product rule is deliberately kept as plain multiplication of the options at each stage. You will sometimes calculate something that is a factorial in disguise — arranging 5 objects is 5 × 4 × 3 × 2 × 1 = 120 — but you are expected to reach it by counting stages, not by using a factorial button or a combinations formula. Those come at A level.
Can I use a sample space diagram instead?
For two stages with a small number of options, yes, and drawing one is a good way to convince yourself the rule works: a 4 × 6 grid visibly has 24 cells. Beyond two stages, or once the numbers grow, a diagram becomes unmanageable and the marks are awarded for the multiplication, so switch to the product rule.
For Socratic GCSE maths practice with Professor Pi — who asks the next question rather than giving the answer — visit aitutors.me.