Permutations and Combinations
Decide whether order changes the outcome before choosing a formula.
Permutations and Combinations
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Order is the deciding question
Use a permutation when rearranging or assigning distinct positions. Use a combination when selecting a group whose order does not matter.
nPᵣ = n!/(n−r)!nCᵣ = n!/[r!(n−r)!]Different roles mean order
President–secretary is different from secretary–president.
Committees ignore order
The same members form one committee regardless of listing order.
Arrange = P; choose = C
“Choose 3 from 8” gives C(8,3) = 56. “Assign three medals among 8” gives 8P3 = 336.
Why it works
A permutation counts each ordering; dividing by r! collapses those orderings into one combination.
Five forms you should recognize
Problem: A canteen offers 3 main dishes and 4 drinks. If a meal contains one main and one drink, how many different meals are possible?
3 × 4 = 12Why: Each main can be paired with any of the four drinks.
Problem: Seven finalists compete for president and vice president. How many officer assignments are possible?
7P2 = 7×6 = 42Why: The positions are different, so reversing two students creates a different assignment.
Problem: A three-person committee is selected from eight students. How many committees are possible?
C(8,3) = 56Why: The same three students form one committee regardless of listing order.
Problem: How many distinct arrangements can be made from the letters of MAMA?
4!/(2!2!) = 6Why: Divide by 2! for the repeated M’s and 2! for the repeated A’s.
Problem: Five students line up, but Ana and Ben must stand together. How many arrangements are possible?
4! × 2 = 48Why: Treat the pair as one block, then count its two internal orders: AB and BA.
Check before you commit
- Using combinations for officer positions
- Using permutations for a committee
- Allowing a leading zero in a number
- Forgetting repeated letters
- Treating circular arrangements as linear
- Counting “at least one” case by case when a complement is faster
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
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Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
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Permutations and Combinations FAQ
When does order matter?
When changing positions, ranks, roles, or sequence creates a different outcome.
Why is circular arrangement (n−1)!?
Rotations of the same seating are equivalent, so fix one person as a reference.
How do I handle “at least one”?
Often count all outcomes and subtract the outcomes with none.
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