Counting Principles

Systematic Lists

What are systematic listing strategies?

Systematic listing strategies are a way of writing out all possible combinations or arrangements of items in an organised way, without missing any
A strategy can be used to make sure that no options are missed out by choosing one item as the first possible option, then arranging the rest using a similar technique
For example, to list all the ways arrange the letters A, B and C
- Begin by fixing the letter A and arranging the others
  A B C or A C B
- Then fix the letter B and arrange the others
  B A C or B C A
- Then fix the letter C and arrange the others
  C A B or C B A
- So the six arrangements are ABC, ACB, BAC, BCA, CAB and CBA
- This idea can be applied to arrangements of more items and longer lists

Exam Tip

Using systematic listing techniques can be particularly helpful in probability, practice finding methods that will ensure you do not miss out any options.

Worked example

List all the possible combinations of one filling, one side and one sauce from the sandwich menu below.

Filling	Side	Sauce
Steak	Lettuce	Ketchup
Chicken	Tomato	Mayo
Veggie Patty	Bacon	Apple

Simplify the problem by assigning a letter to each of the items.

Filling	Side	Sauce
Steak S	Lettuce L	Ketchup K
Chicken C	Tomato T	Mayo Y (M is taken, use a different letter)
Veggie Patty V	Mushroom M	Apple A

‘Fix’ the letter S and the letter L and arrange the other possibilities.

S L K
S L Y
S L A

‘Fix’ the letter S and the letter T and arrange the other possibilities.

S T K
S T Y
S T A

‘Fix’ the letter S and the letter M and arrange the other possibilities.

S M K
S M Y
S M A

Repeat the process by fixing the letter C in the first position.

C L K	C T K	C M K
C L Y	C T Y	C M Y
C L A	C T A	C M A

Repeat the process again by fixing the letter V in the first position.

V L K	V T K	V M K
V L Y	V T Y	V M Y
V L A	V T A	V M A

So there are 27 different ways of choosing one item from each of the options on the sandwich menu, and they can be listed as follows:

S L K, S L Y, S L A	S T K, S T Y, S T A	S M K, S M Y, S M A
C L K, C L Y, C L A	C T K, C T Y, C T A	C M K, C M Y, C M A
V L K, V L Y, V L A	V T K, V T Y, V T A	V M K, V M Y, V M A

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

Exam Tip

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:

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