The first book can be any one of the 5. For each of those . . . The 2nd book can be any one of the remaining 4. For each of those ... The 3rd book can be any one of the remaining 3. For each of those . . . The 4th book can be either of the remaining 2. For each of those . . . The 5th book is the last one remaining.
Total number of ways to arrange the 5 books is
(5 · 4 · 3 · 2 · 1) = 120 .
Cross off alphabetically ascending (1 way), and alphabetically descending (1 way), and you're left with (120 - 2) = 118 ways.
