Count how many outcomes or arrangements are possible by multiplying the number of choices at each stage. It is Higher only, aimed at grade 6. Free questions with answers are on this page. Worked examples show each line before you try the questions.
If a password has four slots and each slot has a number of choices, multiply the choices: that is the product rule. A 3-digit even code using digits 1–9 with no repeat: fill the last (even) slot first, then the others with what is left. Restrictions change a slot's number; they do not change the 'multiply the slots' idea.
When two cases cannot overlap (starts with 1, or starts with 2), work each case and add. When order does not matter you are into combinations, which at GCSE usually still wants a listing or a product divided by the repeats. Write the slots as _ _ _ _ on the page.
Multiply the number of choices at each stage. Reduce by one when repeats are banned. Do a restricted slot (first digit not 0, last digit even) first. Adding the choices is the classic wrong method.
Objectives
Multiply independent choices.
Count ordered pairs when the second choice is smaller.
Count unordered pairs by dividing by 2.
Key points
If one choice has m options and an independent second choice has n options, there are m×n combined outcomes.
Extend to three or more stages by multiplying every stage together.
Reduce the choices at each stage when repeats are not allowed.
Handle a restriction (such as 'the first digit cannot be 0') by dealing with the restricted position first.
Worked examples
Example 1
Nia packs 3 shirts, 2 trousers and 4 hats. Calculate the number of outfits.
2 marks
Identify that the choices are independent.
Multiply the counts together: 3 × 2 × 4 = 24.
The total number of outfits is 24.
Answer: 24
Common slips
Slip: Add the three numbers together: 3 + 2 + 4. Choices from separate categories combine by multiplication, not addition.
Slip: Nia multiplied only shirts and hats: 3 × 4 = 12. Every category must be included in the product.
Slip: Conclude 9 outfits. Adding gives 9 instead of multiplying to 24.
Example 2
Otto picks a captain and a vice-captain from 8 people. Calculate the number of ways.
2 marks
There are 8 choices for captain.
Once the captain is chosen, 7 people remain for vice-captain: 8 × 7 = 56.
The total number of ordered pairs is 56.
Answer: 56
Common slips
Slip: Pick 8 choices for both roles: 8 × 8. One person cannot hold both captain and vice-captain roles.
Slip: Otto wrote 8 × 8 = 64. The second person chosen must be different from the first.
Slip: Divide by 2 to get 28. The roles are distinct (captain and vice-captain), so order matters and we do not divide by 2.
Example 3
Choose 2 people from 5 when order does not matter. Calculate the number of ways.
3 marks
Calculate the ordered pairs: 5 × 4 = 20.
Since order does not matter, each pair AB is identical to BA.
Divide by 2: 20 ÷ 2 = 10.
Answer: 10
Common slips
Slip: Square the number: 5 × 5 = 25. A person cannot be paired with themselves.
Slip: Pia stopped at 20. Leaving the answer as 20 counts every pair twice.
Slip: Divide by 5. There are 2 permutations per pair, so divide by 2.
Questions and answers
GCSE Maths the product rule for counting questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Quin packs 2 shirts, 5 trousers and 3 hats. Calculate the number of outfits.
Show answer
Answer: 30
Multiply independent choices: 2 × 5 = 10.
Multiply by hats: 10 × 3 = 30.
M1 — Multiply independent choices: 2 × 5 = 10.
A1 — Multiply by hats: 10 × 3 = 30.
Question 2 Easy2 marks
Rex packs 4 shirts, 3 trousers and 2 hats. Calculate the number of outfits.
Show answer
Answer: 24
Multiply independent choices: 4 × 3 = 12.
Multiply by hats: 12 × 2 = 24.
M1 — Multiply independent choices: 4 × 3 = 12.
A1 — Multiply by hats: 12 × 2 = 24.
Question 3 Medium2 marks
Uma has 4 choices and 7 choices. Calculate the combinations.
Show answer
Answer: 28
Multiply the two counts.
4 × 7 = 28.
M1 — Multiply the two counts.
A1 — 4 × 7 = 28.
Question 4 Medium2 marks
Val has 8 choices and 2 choices. Calculate the combinations.
Show answer
Answer: 16
Multiply the two counts.
8 × 2 = 16.
M1 — Multiply the two counts.
A1 — 8 × 2 = 16.
Question 5 Hard2 marks
Choose 2 people from 6 when order does not matter. Calculate the number of ways.
Show answer
Answer: 15
Calculate ordered pairs: 6 × 5 = 30.
Divide by 2 for unordered: 30 / 2 = 15.
M1 — Calculate ordered pairs: 6 × 5 = 30.
A1 — Divide by 2 for unordered: 30 / 2 = 15.
Question 6 Hard2 marks
Choose 2 people from 7 when order does not matter. Calculate the number of ways.
Show answer
Answer: 21
Calculate ordered pairs: 7 × 6 = 42.
Divide by 2 for unordered: 42 / 2 = 21.
M1 — Calculate ordered pairs: 7 × 6 = 42.
A1 — Divide by 2 for unordered: 42 / 2 = 21.
Question 7 Exam-style4 marks
A PIN has 4 digits from 0 to 9. No digit is used twice and the first digit cannot be 0. How many different PINs are there?
Show answer
Answer: 4536
The first digit cannot be 0, so it has 9 choices.
The second digit can be 0 but cannot repeat the first, so it has 9 choices.
The third has 8 choices and the fourth has 7.
9 × 9 × 8 × 7 = 4536.
B1 — 9 choices for the first digit
M1 — 9 choices for the second digit
M1 — 8 and 7 choices for the third and fourth digits
A1 — 4536
Common mistakes
Most lost marks on this topic are one slip.
Adding the counts of shirts, trousers and hats
Multiply independent choices
Allowing the same person to be captain and vice-captain
Reduce the second count by 1
Leaving an unordered pair as an ordered product
Divide by 2 when order does not matter
Spot the slip
Mo has 3 shirts, 2 trousers and 4 hats, and multiplies only the shirts and hats. How many outfits?
24
12
9
Show the slip
24 Every shirt can go with every pair of trousers and every hat, so multiply all three.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the multiplication out in full (9×9×8×7) before working it out; the product line is where the method mark sits.
A one-sentence reason ('because each starter can go with each main') can be worth a mark on an 'explain' question.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.