Permutations & Combinations · Grades 12–12
Permutations & Combinations Worksheets
Does the order matter? If it does it's a permutation, if not a combination, and that single question is most of the skill. Worksheets for ₙPᵣ and ₙCᵣ.
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What it is
A factorial n! multiplies every whole number from n down to 1 (5! = 5 × 4 × 3 × 2 × 1 = 120). A permutation ₙPᵣ counts arrangements of r items from n where order matters: ₙPᵣ = n! ÷ (n − r)!. The (n − r)! matches the bottom of n!, so those like factors cancel and only r survive, counting down from n. A combination ₙCᵣ = ₙPᵣ ÷ r! removes the r! repeated orderings.
Why it matters
Counting problems look simple until the order question shows up, and getting that right is the entire skill probability and statistics later lean on. A race podium and a committee can be the same five people, arranged by two completely different rules.
How to do it
- Decide whether order matters (permutation, P) or not (combination, C).
- Write ₙPᵣ as the fraction n! ÷ (n − r)!.
- Cancel the (n − r)! tail shared by top and bottom, then multiply the r numbers left, counting down from n.
- For a combination, divide that result by r!: ₙCᵣ = ₙPᵣ ÷ r!.
Examples by level
Read each question carefully. They're not all the same kind of question.
- BeginnerOrder 3 of 6 songs: P or C?Answerpermutation
- Intermediate₇C₃ =Answer35
- Advanced₁₁C₆ =Answer462
Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.
Common mistakes
- Using a permutation when order doesn't matter (or vice versa).
- Dividing by the wrong factorial in a combination.
- Miscounting the number of factors in ₙPᵣ.
Tips
- Order matters → permutation; order doesn't → combination.
- The (n − r)! tail always cancels, so you never multiply all of n! by hand, and ₙCᵣ is always smaller than ₙPᵣ.
For parents
Ask: does the order matter? If yes, it's a permutation; if no, it's a combination.
For teachers
Anchor the difference with a concrete example, a race podium (order matters) vs. a committee (order doesn't).
Key vocabulary
- factorial (n!)
- the product of every whole number from n down to 1
- permutation
- an arrangement where order matters; ₙPᵣ = n! ÷ (n − r)!
- combination
- a selection where order does not matter; ₙCᵣ = ₙPᵣ ÷ r!
Frequently asked questions
- What is the difference between a permutation and a combination?
- In a permutation the order matters; in a combination it does not. ₙCᵣ = ₙPᵣ ÷ r!.
- How do you calculate ₙPᵣ?
- Multiply r numbers starting from n and counting down: ₅P₃ = 5 × 4 × 3 = 60.
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