Permutations and Combinations
A free Algebra II lesson from the “Sequences, Series, and Counting” unit, with a worked example and practice problems including step-by-step solutions.
A factorial n! multiplies all positive integers from 1 to n. PERMUTATIONS count ordered arrangements: P(n, r) = n! / (n - r)!. COMBINATIONS count unordered selections: C(n, r) = n! / (r! * (n - r)!). Use a permutation when order matters (rankings, lineups) and a combination when it does not (teams, hands of cards).
What you'll learn
- Compute factorials (n!) and use them in counting formulas
- Use P(n, r) = n! / (n - r)! for ordered arrangements
- Use C(n, r) = n! / (r! * (n - r)!) for unordered selections
- Decide whether a problem calls for a permutation (order matters) or a combination (order does not)
Worked example
Problem. How many 3-letter ordered codes can you make from the letters A, B, C, D, E with no repeats?
- Order matters and no repeats -> permutation P(5, 3).
- P(5, 3) = 5 * 4 * 3 = 60.
Answer: 60
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response). Counts include repeated prompts and variants.
Practice problems
1. Compute 5!.
Show solution
- 5 * 4 * 3 * 2 * 1 = 120.
Answer: 120
2. Compute 4!.
Show solution
- 4 * 3 * 2 * 1 = 24.
Answer: 24
3. Compute 3!.
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- 3 * 2 * 1 = 6.
Answer: 6
4. Compute P(5, 3).
Show solution
- P(5, 3) = 5 * 4 * 3 = 60.
Answer: 60
5. Compute P(6, 2).
Show solution
- P(6, 2) = 6 * 5 = 30.
Answer: 30
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