In how many ways can (5) books be selected from (8) distinct books and arranged on a shelf?

Correct answer: (6720)

The direct answer is option A, 6720. The words “selected and arranged on a shelf” mean that both choosing and ordering matter. First choose 5 books from 8 and then arrange those 5 books: \(\binom{8}{5}\times5! = \frac{8!}{3!}=8P5=6720\). Equivalently, fill the five shelf positions successively: 8 choices for the first, 7 for the second, 6 for the third, 5 for the fourth, and 4 for the fifth, giving \(8\times7\times6\times5\times4=6720\). Option A is correct. Option B, 336, is \(8P?\) not the required complete count and does not represent selecting and ordering five books correctly; in fact \(\binom{8}{3}=56\), while \(8P5=6720\). Option C, 56, is \(\binom{8}{5}\), so it counts selection only and ignores shelf order. Option D, 40320, is \(8!\), which arranges all 8 books, whereas only 5 books are to be placed. Exam cue: if selected objects are also placed in positions, use permutation, not combination.