LEARNING OBJECTIVES
What you will be able to do
- use factorial notation including 0!
- calculate permutations and combinations
- select the correct method from context
- solve arrangement and selection problems within syllabus limits
AT A GLANCE
INTRODUCTION · THE BIG IDEA
Translate wording into a counting model by deciding whether order changes the outcome.
Counting problems become manageable when each choice is described once and without overlap. The central decision is whether swapping selected objects creates a new outcome.
The 0606 syllabus uses distinct objects without repetition and excludes circular arrangements and problems mixing permutation and combination stages.
SECTION 01
Factorials and the multiplication principle
The multiplication principle multiplies the number of choices at consecutive stages. Factorial notation packages a descending product compactly.
| Included | Not included |
|---|---|
| distinct objects | repeated identical objects |
| linear arrangements | objects arranged in a circle |
| one permutation or combination model | problems requiring both models |
SECTION 02
Permutations: order matters
A permutation counts ordered arrangements of r objects selected from n distinct objects. The first position has n choices, the next n−1, and so on.
Form an ordered code
- Choose a 4-character code from 7 distinct symbols without repetition.
- Positions matter, so use a permutation.
- Compute ⁷P₄ = 7 × 6 × 5 × 4.
Answer: 840 codes.
SECTION 03
Combinations: order does not matter
A combination counts selections where rearranging the same chosen objects makes no new outcome. Dividing by r! removes the internal orders of each group.
DETAILED EXPLANATION
- Choosing r included objects is equivalent to choosing n−r excluded objects.
- The answer must be a non-negative integer.
- Simplify factorials before multiplying large numbers.
SECTION 04
Restrictions and algebraic counting
Handle a condition by fixing required objects, excluding forbidden ones or separating mutually exclusive cases. Algebraic questions may ask for n or r from an equation involving nPᵣ or nCᵣ.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- Factorials represent descending products and 0! = 1.
- Permutations count ordered selections.
- Combinations count unordered selections.
- Restrictions should be handled before counting.
- Circular and repeated-object arrangements are outside this syllabus.
QUICK REVISION CHECKLIST
Can you do each of these without your notes?
- use factorial notation including 0!
- calculate permutations and combinations
- select the correct method from context
- solve arrangement and selection problems within syllabus limits