Arrange 5 different books in a row

How many different arrangements are there for placing 5 distinct books in a row on a bookshelf?

Step-by-step

1. Total n = 5, chosen r = 5
   First distinguish the total number from the chosen number, then check whether order matters.
2. Permutations A(5, 5) = 5! / (5 − 5)! = 120
   Permutations depend on order, so we compute by dividing factorials.
3. Combinations C(5, 5) = 5! / (5!(5 − 5)!) = 1
   Combinations do not depend on order; compared to permutations, we divide by an extra r!.
4. Order matters ⇒ use permutations: 120
   Deciding whether order matters is the most crucial step in this type of problem.

Why it works

Arranging distinct books in a row depends on order, because swapping any two books produces a new arrangement, so we must use permutations. When all are placed, the number of arrangements equals the factorial of the count, meaning multiply from the last position downward one by one. Determining whether order matters is the key to this type of problem; confirm first before writing.

Common mistakes

• Calculated permutations as combinations, omitting the factorial of the number taken out.
• Confused the number of books with the number of positions, using the wrong factorial.
• Assumed "arranged in a row" is independent of order and directly used combinations.

Related

n-math-counting-combination