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Systematic Listing Strategies
Select the skills to practice, and then click Go!
Foundational skills
Identify categories
\[ \text{Shirts} \times \text{Trousers} \]
Recognise how many different groups you are choosing from.
Recognise whether order matters
\[ \text{PIN Code vs Team Selection} \]
Decide whether the order of selection makes a difference.
Two-category combinations
List combinations: Small
\[ \text{A1, A2, B1, B2} \]
List every possible combination (4-6 outcomes).
List combinations: Medium
\[ \text{A1, A2… B1, B2…} \]
List every possible combination (8-12 outcomes).
Calculate combinations
\[ 3 \times 4 = 12 \]
Use the product rule to calculate total combinations.
Find missing
\[ \text{List: A1, A2, B1…} \]
Find the one combination missing from a list.
Three-category combinations
List 3 categories
\[ \text{A1x, A1y, A2x…} \]
List every combination from three categories.
Calculate 3 categories
\[ 2 \times 3 \times 2 = 12 \]
Use the product rule for three categories.
Find missing (3 cats)
\[ \text{A1x, A1y, A2x…} \]
Find the missing combination from a list.
Arrangements: All items
List arrangements (3 items)
\[ \text{ABC, ACB, BAC…} \]
List all 6 arrangements of 3 items.
Calculate arrangements
\[ 4! = 24 \]
Calculate arrangements of 3 or 4 items.
Arrangements: Selecting items
List arrangements: 2 from 3
\[ \text{AB, AC, BA, BC…} \]
Order matters. Select 2 from 3.
List arrangements: 2 from 4
\[ \text{12, 13, 14, 21…} \]
Order matters. Select 2 from 4.
Calculate selections
\[ 5 \times 4 = 20 \]
Calculate arrangements when selecting from a larger set.
Codes and PINs
List codes: 2 pos (repeat)
\[ \text{11, 12, 13, 21…} \]
List codes where digits can repeat.
List codes: 2 pos (no repeat)
\[ \text{AB, AC, AD…} \]
List codes where letters cannot repeat.
Calculate 2-position codes
\[ 4 \times 4 = 16 \]
Calculate total codes with/without repetition.
Calculate 3-position codes
\[ 5 \times 4 \times 3 = 60 \]
Calculate total 3-digit PINs.
Selections: Order doesn’t matter
List selections: 2 from 4
\[ \text{AB, AC, AD, BC…} \]
List ways to choose 2 from 4 (order irrelevant).
List selections: 2 from 5
\[ \text{AB, AC… DE} \]
List ways to choose 2 from 5 (order irrelevant).
Find missing selection
\[ \text{AB, AC… (missing)} \]
Find the missing pair from a list.
Calculate selections
\[ \text{Choose 2 from 5} = 10 \]
Calculate number of ways to choose items.
Special cases
Combinations with restrictions
\[ 9 – 1 = 8 \]
Calculate combinations when some pairings are not allowed.
Larger product rule
\[ 5 \times 4 \times 3 = 60 \]
Calculate combinations for larger sets.
Timer (Optional)
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Question