The product rule for counting

The product rule for counting says that if one choice can be made in m ways and a second, separate choice in n ways, then the two together can be made in m × n ways. With more choices, keep multiplying. A café with 4 sandwiches and 3 drinks offers 4 × 3 = 12 different meal deals. In the US it is called the fundamental counting principle.

What is the product rule for counting?

Counting by listing works for small numbers, but it quickly gets out of hand. The product rule lets you count every combination without writing them all down. Picture each choice as a box, write how many options go in each box, and multiply.

Here n₁ is the number of options for the first choice, n₂ for the second, and so on. It only works when every option at one step can go with every option at the next.

Why it works

Take 4 shirts and 3 pairs of trousers. Each of the 4 shirts can be worn with any of the 3 pairs, so each shirt gives 3 outfits, and 4 shirts give 4 × 3 = 12. A grid with shirts down the side and trousers along the top has 4 rows and 3 columns: 12 cells, one for each outfit. Add 2 pairs of shoes and every one of those 12 outfits splits into 2, giving 24.

How to use the product rule for counting

1. Split the task into steps: one for each position, digit, letter or choice.
2. Draw a box for each step.
3. Fill in the number of options for each box. Deal with any restrictions first.
4. Multiply the numbers in the boxes.

Example 1: a PIN code

How many 4-digit PIN codes are there, if any digit 0 to 9 can be used in each place, with repeats allowed?

Each of the 4 boxes has 10 options:

Example 2: no repeats allowed

A 3-digit code uses the digits 0 to 9, and no digit can be used twice. How many codes are possible?

The first digit has 10 options. Once it is used, the second has 9, and the third has 8.

Example 3: arranging all the items

In how many different orders can 5 people stand in a line?

5 choices for the first place, then 4, 3, 2 and 1: 5 × 4 × 3 × 2 × 1 = 120. This product is written 5! and called 5 factorial.

Example 4: choosing for different jobs

A team of 15 players chooses a captain and a vice-captain. How many ways can this be done?

15 choices for captain, then 14 left for vice-captain: 15 × 14 = 210. The jobs are different, so the order matters.

Product rule questions with restrictions

When some positions have rules, fill those boxes first, then the rest.

Example 5: even numbers

How many 3-digit even numbers can be made from the digits 1, 2, 3, 4, 5 and 6, if no digit is repeated?

The number is even if its last digit is even, so start there: 2, 4 or 6 gives 3 options. That leaves 5 digits for the first place and 4 for the middle.

Example 6: greater than a value

How many 4-digit numbers greater than 5000 can be made from the digits 1 to 9, with repeats allowed?

The first digit must be 5, 6, 7, 8 or 9: 5 options. Each of the other three digits has 9 options.

Example 7: number plates

A design for number plates uses 2 letters followed by 3 digits, with repeats allowed. How many plates are possible?

26 × 26 × 10 × 10 × 10 = 676 000.

Now try listing outcomes yourself. Find the sample space, then press Mark to check it in the course.

The product rule and the sum rule

The product rule is for choices made one after another, where you do one thing and another. If you do one thing or another, add instead.

• A café has 4 sandwiches and 3 wraps. Choosing one sandwich and one wrap: 4 × 3 = 12 ways.
• Choosing a sandwich or a wrap, just one item: 4 + 3 = 7 ways.

The same AND and OR pattern runs through probability, where AND means multiply and OR means add for mutually exclusive events.

Product rule, permutations and combinations

Examples 2 to 4 are permutations: arrangements where order matters. When order does not matter, such as choosing 2 people for a committee rather than for named jobs, every pair has been counted twice, so divide by the number of orders. Choosing 2 from 15 with no named roles gives 210 ÷ 2 = 105 ways. A-level and US courses build this into the formulas for nPr and nCr. The Permutations and Combinations Calculator does these calculations for you.

Common mistakes

• Adding instead of multiplying. One choice and another means multiply.
• Forgetting the number goes down without repeats. If each item can only be used once, each box has one fewer option than the last.
• Filling restricted boxes last. Deal with the restrictions first, such as an even last digit or a first digit that cannot be 0.
• Counting arrangements when order does not matter. If swapping two items gives the same thing, divide by the number of orders.
• Allowing 0 as a first digit when the question asks for a 3-digit or 4-digit number.

The product rule for counting at GCSE

AQA's GCSE Maths specification includes it in the number topic N5: "apply systematic listing strategies including use of the product rule for counting", where systematic listing includes lists, tables and diagrams. Questions often ask for the number of possible codes, outfits or arrangements, and sometimes ask you to show how you know. The Probability course covers listing outcomes and sample spaces, which lead into this.

Practise in a short timed paper

GCSE Maths Foundation and GCSE Maths Higher each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the whole course.

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