ECON1202 Chap.4 Counting, Permutations and Combinations
Counting, Permutations and Combinations
Counting begins by deciding what makes two outcomes different. The multiplication principle joins sequential stages when one complete outcome chooses an option at every stage. If selections remove future options, the counts fall; when repetition is allowed, a stage count may remain constant. Permutations count ordered selections because exchanging selected objects between named positions creates a new outcome.
Combinations count subsets because internal order does not matter, so the ordered count is divided by the factorial number of arrangements inside each chosen group. Restrictions are handled by adding mutually exclusive cases or subtracting a simpler complement. Integer-allocation questions use stars and bars, with different placement rules for non-negative and positive parts.
Circular arrangements remove rotations by fixing a reference object, while reflections remain different unless the question identifies them. A robust solution states the equivalence rule in words, shows whether order and repetition matter, and separates multiplication within a case from addition across cases. Bounds against related counts provide a quick verification after the arithmetic.
What this chapter covers
- 01
Multiplication principle
- 02
Ordered selections
- 03
Permutations without repetition
- 04
Unordered subsets
- 05
Combinations and symmetry
- 06
Restricted-case addition
- 07
Stars and bars
- 08
Circular arrangements
Worked example · free
Select an ordered leadership team
- +1The roles are named, so assigning the same people to different roles creates a different outcome.
- +1There are ten choices for president, nine remaining for secretary and eight for treasurer.
- +1Multiply the stages: 10 x 9 x 8 = 720.
- +1Check that this exceeds the unordered count 10C3 because every selected trio has six role assignments.
Key terms
- Multiplication principle
- Rule that multiplies the number of available choices across successive stages of one outcome.
- Permutation
- An ordered selection of objects from a larger set.
- Combination
- An unordered subset selected from a larger set.
- Mutually exclusive cases
- Separate cases that cannot describe the same outcome and can therefore be added.
- Complement count
- The total count minus the outcomes that violate a stated condition.
- Circular permutation
- An arrangement around a circle where rotations represent the same outcome.
Counting, Permutations and Combinations FAQ
How do I decide between a permutation and a combination?
Ask whether swapping two selected objects creates a different outcome. Named jobs, ranks and positions make order matter, so use a permutation. A committee with identical roles depends only on membership, so use a combination.
When should counts be multiplied or added?
Multiply choices that occur as successive stages inside one complete outcome. Add counts for mutually exclusive cases that reach the requested condition in different ways. State the cases so duplicate outcomes cannot appear in two totals.
How do repetition restrictions change the count?
With repetition allowed, each stage may retain the same number of choices. Without repetition, later stages lose previously selected objects. For restricted repetition, describe the legal choices at each stage or divide into clean cases.
What is the quickest check on a counting answer?
Compare it with nearby structures. An ordered selection should be at least as large as the corresponding combination and no larger than the count that permits unrestricted repetition. Also verify the result is an integer and every case has a distinct interpretation.
How should I justify a counting method in an exam response?
Describe the outcome being counted, then state whether order matters, repetition is allowed and any position or category is restricted. For a product count, identify what each factor chooses and how the available set changes. For a combination, explain why rearranging the selected objects does not create a new outcome; if roles are assigned later, count that ordered stage separately.
When using complementary counting, define the full set and the excluded cases and check that exclusions do not overlap unexpectedly. For circular arrangements, state whether rotations and reflections are considered identical. For stars and bars, state what the stars and separators represent and whether zero allocations are permitted.
The final integer should be compared with an unrestricted upper bound and a simpler nearby case.
Exam move
Before using a formula, write three labels: order matters or not, repetition allowed or not, and restrictions. Build simple counts through stage boxes and translate the product into a sentence. Pair each permutation question with an unordered version so the role of the dividing r factorial becomes visible. For at-least conditions, practise both disjoint cases and complements, then choose the shorter argument.
Draw stars and separators for small integer allocations before memorising the formula; modify the representation when every part must be positive. For circles, fix one object and arrange the rest, then decide whether reflections are distinct. Verify with a smaller version that can be listed by hand. The aim is to make the equivalence rule explicit so factorial notation records the structure rather than replacing it.
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