What is meant by counting principles?

• The fundamental counting principle states 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…
  • … analyse patterns and make generalisations about real work situations
  • … find the number of?permutations?of?n?items
  • … find the number of ways of choosing an item from a list of?n?items
  • … find the number of ways of choosing?r?items from?n?items
  • … find the number of ways of permutating?r?items from?n?items
• The topic of counting principles is a particularly interesting part of mathematics that can lead to the development of working with very large numbers
• It is always vital to consider whether objects taken from each list can be repeated or not
  • For example a four digit PIN from ten numbers where each number can be used repeatedly would be 10?×?10?×?10?×?10
    • There are 10 options for the first and ten options for the second number and so on
  • If the numbers could only be used once then the number of options for each digit would?reduce?with each digit
    • There are 10 options for the first, nine options for the second, eight for the third and so on
    • This concept will be explored further in the?permutations?revision note

How do I choose an item from a list of?m?items AND another item from a list of?n?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?m?items OR another item from a list of?n?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