Counting Principles (Edexcel GCSE Maths: Foundation)

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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

Examiner 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

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.