2022 Cayley Problem 14

The integer 48\,178 includes the block of digits 178. The three integers 51\,870, 19\,728 and 38\,717 do not include the block of digits 178. How many integers between 10\,000 and 100\,000 include the block of digits 178?

\textbf{(A)}\ 280\quad \textbf{(B)}\ 300\quad \textbf{(C)}\ 270\quad \textbf{(D)}\ 310\quad \textbf{(E)}\ 260

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.