1. **State the problem:** We want to find the biggest sum of the digits displayed on a 24-hour digital clock in the format HH:MM. 2. **Understand the clock format:** The clock shows hours from 00 to 23 and minutes from 00 to 59. 3. **Sum of digits formula:** For a time HH:MM, the sum is $H_1 + H_2 + M_1 + M_2$ where $H_1$ and $H_2$ are the digits of the hour, and $M_1$ and $M_2$ are the digits of the minutes. 4. **Maximize the sum:** - The hour digits can be from 00 to 23. - The minute digits can be from 00 to 59. 5. **Check the maximum digit sums for hours:** - Hours 00 to 09: max sum is $0 + 9 = 9$ (at 09) - Hours 10 to 19: max sum is $1 + 9 = 10$ (at 19) - Hours 20 to 23: max sum is $2 + 3 = 5$ (at 23) So the maximum hour digit sum is 10 at 19. 6. **Check the maximum digit sums for minutes:** - Minutes range from 00 to 59. - Max digit sum is $5 + 9 = 14$ (at 59). 7. **Calculate the total maximum sum:** $$ \text{max sum} = 10 + 14 = 24 $$ 8. **Conclusion:** The biggest sum Romeo can get from his 24-hour clock is **24**. **Final answer:** (c) 24