From page 1 to page 9 have 9 digits

From page 10 to page 99 have:

\(\left(99-10+1\right)\times2=180\) [digits]

From page 100 to page 120 have:

\(\left(120-100+1\right)\times3=63\) [digits]

To number all pages that need the number of digits are:

9 + 180 + 63 = 252 [digits]

Answer: 252 digits