The product rule for counting says that if one choice can be made in m ways and a second in n ways, the two together can be made in m × n ways. It extends to any number of stages. When items cannot be reused, the number of choices drops by one at each stage, and when there is a restriction, deal with the restricted position first.
Counting rules
- Product rule: m ways then n ways gives m × n ways
- Repeats allowed: k choices for each of r positions gives kʳ
- No repeats: n × (n − 1) × (n − 2) × …
- n different items in a line: n × (n − 1) × … × 1 = n!
- Restrictions: fill the restricted position first
The product rule
If a code is one letter followed by one digit, there are 26 choices for the letter and 10 for the digit, so 26 × 10 = 260 codes. Every choice at one stage can be combined with every choice at the next, which is why you multiply rather than add. The rule extends to any number of stages.
Repeats allowed
When each position can use any of the same k options, multiply k by itself once for each position. A password of three letters, where letters can repeat, has 26 × 26 × 26 = 26³ = 17 576 possibilities.
Arrangements without repetition
When each item can be used only once, the number of choices falls by one at each stage. Arranging 6 people in a queue gives 6 choices for the front, then 5, then 4, and so on: 6 × 5 × 4 × 3 × 2 × 1 = 720. This product is written 6! and read as “6 factorial”.
Counting with restrictions
If one position has a restriction, fill it first, then fill the others with whatever is left. For three-digit even numbers made from 1, 2, 3 and 4 with no repeats, the last digit must be 2 or 4, giving 2 choices; then 3 choices remain for the first digit and 2 for the middle, so 2 × 3 × 2 = 12.
Choosing for different roles
Selecting people for different named roles, such as captain and vice-captain, is an arrangement, because the order matters. From 8 players there are 8 choices for captain and then 7 for vice-captain, so 8 × 7 = 56 ways.
Worked examples
Example 1. A password is 3 letters (A–Z), and letters may repeat. How many passwords are possible?
- 26 choices for each of 3 positions.
- 26 × 26 × 26.
Answer: 17 576.
Example 2. In how many ways can 6 people stand in a queue?
- 6 choices, then 5, then 4, 3, 2, 1.
- 6 × 5 × 4 × 3 × 2 × 1.
Answer: 720 ways.
Example 3. How many three-digit even numbers can be made from 1, 2, 3 and 4 with no digit repeated?
- Last digit first: it must be 2 or 4, so 2 choices.
- Then 3 choices for the first digit and 2 for the middle.
Answer: 2 × 3 × 2 = 12.
Example 4. A licence code is 2 letters followed by 2 digits, with repeats allowed. How many codes are possible?
- Letters: 26 × 26. Digits: 10 × 10.
- 26² × 10².
Answer: 67 600.
Example 5. A team of 8 must choose a captain and a vice-captain. In how many ways can this be done?
- 8 choices for captain.
- 7 choices for vice-captain from those left.
Answer: 56 ways.
Common mistakes
- Adding instead of multiplying. 3 starters and 5 mains give 3 × 5 meals.
- Ignoring “no repeats”. Reduce the choices by one at each stage.
- Filling restricted positions last. Deal with the restricted position first.
- Counting 26 + 26 for two letters. Two letter positions give 26 × 26.
- Using 10 choices for a first digit that cannot be 0. Check whether every digit is allowed in each position.
How the product rule is examined
The product rule for counting is Higher tier content in Edexcel GCSE (1MA1); simpler versions of the same idea appear in International GCSE (4MA1) listing questions. You may be asked to:
- count codes, passwords or number plates, with or without repeats
- count arrangements of different objects in a line
- count numbers or arrangements that satisfy a restriction
- count ways to fill different roles
Counting skills are built up on the systematic listing worksheet, which has answers. For harder product-rule questions, use the GCSE Number engine.
This extends systematic listing, and counting outcomes is the basis of probability from sample spaces.
Frequently asked questions
What is the product rule for counting?
If one event can happen in m ways and another in n ways, together they can happen in m × n ways.
What does 5! mean?
5 factorial: 5 × 4 × 3 × 2 × 1 = 120, the number of ways to arrange 5 different items in a line.
How do you count with a restriction?
Fill the restricted position first, then multiply by the number of choices left for the other positions.