№13 · NUMBER · FOUNDATION + HIGHER
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gcsemathsai
EST. MMXXIV · LONDON · MMXXVI SPEC.
№ 13 · XIII
NUMBER · FOUNDATION + HIGHER A4 · 210×297mm
NumberFoundation + Higher
Product Rule for Counting.
If one task has m ways and another has n, together there are m × n ways. This is the multiplication principle — the foundation of counting problems, PIN codes, menus, and probability trees.
I · Key definitions
Product rule
If task A has m ways and task B has n ways, total = m × n.
4 shirts × 3 trousers = 12 outfits
Independent
One choice doesn't affect the other.
colour of car & colour of shoes
With replacement
Choose, put back. Same number of choices each time.
3 letters × 3 letters = 9
Without replacement
Each choice reduces the remaining pool.
2 from 5 = 5 × 4 = 20
Restrictions
Handle restricted slots first, then fill rest.
e.g. 'must start with 1'
! (factorial)
n! = n × (n−1) × … × 1. Useful but not required at GCSE.
3 letters from 26 (no repeats): 26 × 25 × 24 = 15,600
No letters repeated, etc.
Apply constraint at each step
'no same letters next to each other' — think carefully
IV · Worked examples
Three-letter codes (with repeats)
2 marks
How many 3-letter codes can be made from the 26 letters of the alphabet, if letters can be repeated?
i.Each slot has 26 choices (repeat allowed)
ii.Total = 26 × 26 × 26 = 26³
iii.= 17,576
17,576 codes
4-digit PINs with restriction
3 marks
How many 4-digit PINs (digits 0–9) are possible if the first digit must be non-zero?
i.Slot 1: can't be 0 → 9 choices (1–9)
ii.Slots 2, 3, 4: any digit → 10 each
iii.Total = 9 × 10 × 10 × 10 = 9000
9,000 PINs
Without replacement
2 marks
A password is a sequence of 3 different letters chosen from A–Z. How many possibilities?
i.Slot 1: 26 choices
ii.Slot 2: 25 (one used)
iii.Slot 3: 24
iv.Total = 26 × 25 × 24 = 15,600
15,600 passwords
V · Common mistakes & examiner tips
Common mistakes
Adding instead of multiplying. 3 + 4 = 7 is wrong for 'choose one of each'.
Allowing repeats when forbidden (or vice versa).
Starting without restriction, then ignoring it. Handle restrictions first.
Forgetting 0 is a digit. 'Digits 0–9' = 10 options, not 9.
Treating different-looking cases as distinct when they're not.
Examiner tips
Draw out the slots — like 4 boxes for a 4-digit code.
Write the number of choices in each slot before multiplying.
Always handle restricted slots first.
Ask: with or without replacement? Crucial for each problem.
For impossible constraints (e.g., 4 letters from 3): answer is 0.
SQUARE · 900 × auto · Social-ready
Σ
gcsemathsai
EST. MMXXIV · LONDON · MMXXVI SPEC.
№ 13 · XIII
NUMBER · FOUNDATION + HIGHER Square · 1:1
NumberFoundation + Higher
Product Rule for Counting.
If one task has m ways and another has n, together there are m × n ways. This is the multiplication principle — the foundation of counting problems, PIN codes, menus, and probability trees.
I · Key definitions
Product rule
If task A has m ways and task B has n ways, total = m × n.
4 shirts × 3 trousers = 12 outfits
Independent
One choice doesn't affect the other.
colour of car & colour of shoes
With replacement
Choose, put back. Same number of choices each time.
3 letters × 3 letters = 9
Without replacement
Each choice reduces the remaining pool.
2 from 5 = 5 × 4 = 20
Restrictions
Handle restricted slots first, then fill rest.
e.g. 'must start with 1'
! (factorial)
n! = n × (n−1) × … × 1. Useful but not required at GCSE.