Quick answer: For nonnegative integers, n! = n(n − 1)…1 and 0! = 1. Arranging r distinct items chosen from n without repetition gives nPᵣ = n!/(n − r)!, because order matters.
Worked example
How many ways can three of five students take first, second and third places?
- There are 5 choices for first place.
- After one student is used, 4 choices remain for second.
- Then 3 choices remain for third.
- Total = 5 × 4 × 3 = 60.
Check the result
5!/2! = 120/2 = 60, matching the slot-counting method.
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Common mistake to avoid
Do not divide by 3! for ranked places: swapping first and second changes the outcome. Repeated items need a different count.
Try it yourself
1. Arrange four different books in a row. 2. Calculate 6!/4!.
Answers and reasoning
1. 4! = 24 arrangements. 2. Cancel 4! to leave 6 × 5 = 30.
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