Problem Solving with Permutations & Combinations (Cambridge O Level Additional Maths)

How will I know whether to use Permutations or Combinations?

• In permutation problems, the order of arrangement matters
  • For example, guessing who's going to win first, second and third prize in a competition
• In combination problems, the order of arrangement doesn't matter
  • For example, guessing the top three high scorers from a group of students
• Look out for keywords in a problem 
  • Permutation questions will most likely use the words 'arrange', 'order', 'sequence'
  • Combination questions will most likely use the words 'select', 'choose', 'group'
• Pay close attention to the context of the problem to determine if the order matters or not

How should I approach perms and combs questions?

• Identify whether the question is a permutations or a combinations question
• Identify the total number of options available (n) and the number to be arranged or selected (r)
• 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
• Look out for any limitations or conditions in the problem 
  • Do some items need to be kept together or separate?
  • Does a code or a number need to begin or end in a particular value? 

How will I know whether to add or multiply?

• 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
• 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
• If a question has a restriction, such as 'four items containing at least one yellow and no more than three reds' you will need to 
  • Find all of the possibilities  (one yellow and three reds or two yellows and two reds etc)
  • Multiply for each of the 'and' statements (one yellow and three reds)
  • Add together each of the 'or' statements at the end (one yellow and three reds or two yellows and two reds etc)