What is generally meant by distinct line spectrum?
Your intuition is right: all the transitions are possible.

But the language is a little vague and might just be a simplification.

The emission spectrum of hydrogen atoms contains a lot of lines which are often grouped into series (which are named Lyman, Ballmer and Paschen, for example). What groups the series is the lower energy level in the transition responsible for the line: the Lyman series are transitions to the lowest energy level; Ballmer is transitions to the second lowest level...). A good summary of all this is given here.

Your book may have been assuming you only counted the lines corresponding to the final drop to the ground state (the Lyman series) ignoring the intermediate drops to other energy levels. If not, it is clearly wrong as all the transitions are possible.