Quick answer: Use a combination when only the selected group matters; use a permutation when positions or roles matter. Without repetition, nCᵣ = n!/[r!(n − r)!], and nPᵣ = r! × nCᵣ.
Worked example
From six students, choose a team of two, then compare choosing a captain and vice-captain.
- An unordered team has 6C₂ = 6 × 5/2 = 15 possibilities.
- For named roles, there are 6 choices for captain and 5 for vice-captain.
- The ordered count is 6P₂ = 30.
Check the result
Each two-person team gives two role assignments, so 15 × 2 = 30.
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Common mistake to avoid
The words choose or select alone are not decisive: check whether assigning different roles changes the outcome.
Try it yourself
1. Choose three books from five. 2. Award gold and silver to two of five runners.
Answers and reasoning
1. 5C₃ = 10 groups. 2. 5P₂ = 5 × 4 = 20 ordered results, assuming no ties.
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