Product Rule for Counting (Edexcel GCSE Maths)
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Product Rule for Counting
What is meant by counting principles?
Counting principles state that if there are m ways to do one thing and n ways to do another there are m × n ways to do both things
Applying counting principles allows us to ...
... see patterns in real world situations
... find the number of combinations or arrangements of a number of items
... find the number of ways of choosing some items from a list of items
When you have a question like “How many ways…?” you should always look for the words “AND” and “OR”
“AND means ×”
“OR means +”
How do I choose an item from a list of items AND another item from a different list of items?
If a question requires you to choose an item from one list AND an item from another list you should multiply the number of options in each list
In general if you see the word 'AND' you will most likely need to 'MULTIPLY'
For example if you are choosing a pen and a pencil from 4 pens and 5 pencils:
You can choose 1 item from 4 pens AND 1 item from 5 pencils
You will have 4 × 5 different options to choose from
How do I choose an item from a list of items OR another item from a different list of items?
If a question requires you to choose an item from one list OR an item from another list you should add the number of options in each list
In general if you see the word 'OR' you will most likely need to 'ADD'
For example if you are choosing a pen or a pencil from 4 pens and 5 pencils:
You can choose 1 item from 4 pens OR 1 item from 5 pencils
You will have 4 + 5 different options to choose from
Examiner Tips and Tricks
Always read a question carefully and identify where it requires you to add or multiply before beginning the problem
Worked Example
Harry is going to a formal event and is choosing what accessories to add to his outfit. He has seven different ties, four different bow ties and five different pairs of cufflinks. How many different ways can Harry get ready if he chooses:
i) Either a tie, a bow tie or a pair of cufflinks?
Harry has 7 + 4 + 5 different items to choose from.
He wants a tie OR a bow tie OR a pair of cufflinks.
OR means add them.
7 + 4 + 5 = 16
16 different ways to choose either a tie, a bow tie or a pair of cufflinks
ii) A pair of cufflinks and either a tie or a bow tie?
Harry wants a tie AND a pair of cufflinks OR a bow tie AND a pair of cufflinks.
AND means multiply them.
A tie AND cufflinks: 7 × 5 = 35
A bow tie AND cufflinks: 4 × 5 = 20
OR means add them.
35 + 20 = 55 ways
55 different ways to choose pair of cufflinks and either a tie or a bow tie
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