Q.In how many ways can 5 different books be arranged on a shelf?

Arranging ALL of a set of distinct objects in a row uses every position, so this is $^{5}P_{5} = 5!$, the total number of orderings.

Since ALL 5 books must be placed on the shelf and their order clearly matters (a different order is a different arrangement of the shelf), this is $^{5}P_{5}$:

This is simply $5!$ itself, exactly as the boundary-case note in this chapter's Permutations section states ($^{n}P_{n}=n!$). …

Reaching for a combination ($^{5}C_{r}$) instead of a permutation, forgetting that a bookshelf arrangement is inherently ordered (swapping two books' positions genuinely changes the shelf). Also, mis-simplifying $5!/0!$ by treat …