Problem Solving with Permutations & Combinations

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)

Examiner Tip

Read the question twice, taking note of the key areas outlined in this revision note and looking out for any keywords
If you're unsure, try writing down a few of the possible options to look for clues

Worked example

A password must be 5 characters long and consist of 2 letters from the letters A, B, C, D, and E followed by 3 digits from the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each letter or number may only be used once within the password. Find how many ways the password can be formed.

There are no keywords to help us here so we must consider to context of the question.
The context is about a password so the order of the arrangement matters, this is a permutations question.

The 2 letters must be followed by the three digits, so we must treat these separately.

For the first letter there are five possible options (A, B, C, D or E)
For the second letter there are four possible options as no letter can be repeated.

${}^{5}P_{2} = (5 \times 4)$

For the first number there are ten possible options (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
For the second number there are nine possible options as no number can be repeated.
For the third number there are eight possible options.

${}^{10}P_{3} = (10 \times 9 \times 8)$

We want two letters AND three numbers so we will multiply the options.

${}^{5}P_{2} \times {}^{10}P_{3}$

14400 passwords

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

Revision Note