B5. My 24-hour digital clock displays hours and minutes only.
For how many different times does the display contain at least one occurrence of the digit 5 in a 24-hour period?
SolUTion
There are 24 hours in any given 24 -hour period.
There are only 2 hours in the period with a ' 5 ' in the hours display (05:00-05:59 and 15:0015:59). During these times, every minute shows a ' 5 ' in the display and, therefore, there are 60 times to count per hour.
In the remaining 22 hours, in each hour at ' 05 ', ' 15 ', ' 25 ', ' 35 ', ' 45 ' and ' 50 ' to ' 59 ' minutes past the hour, there is at least one occurrence of a ' 5 ' and hence there are 15 times in each of these 22 hours where at least one digit 5 occurs.

Therefore, the total number of different times displaying at least one occurrence of the digit ' 5 ' is
$$
2 \times 60+22 \times 15=450 \text {. }
$$