IB DP Maths: AA HL复习笔记1.7.1 Counting Principles

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