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
- Begin by fixing the letter A and arranging the others
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 |
