9.1. The electronic clock on the building of the station shows the current hours and minutes in the format HH:MM from 00:00 to 23:59. How much time in one day will the clock display four different digits?
Answer: 10 hours 44 minutes.

Solution. Each possible combination of four digits burns on the clock for one minute. Consider separately the time of day from 00:00 to 19:59 and from 20:00 to 23:59. In the first case, the number of valid combinations according to the problem's conditions will be: 2 (the tens digit of the hour in the first position), multiplied by 5 (the tens digit of the minute in the third position, except for the one already burning in the first position), multiplied by 8 (the units digit of the hour, not equal to the digits in the 1st and 3rd positions) and multiplied by 7 (the units digit of the minute in the fourth position, not equal to the digits in the 1st, 2nd, and 3rd positions). This results in 560 combinations, burning for 560 minutes, which is 9 hours and 20 minutes per day.

In the second case, the number of valid combinations according to the problem's conditions will be: 1 (the tens digit of the hour in the first position, equal to 2), multiplied by 3 (the units digit of the hour in the second position, not equal to 2), multiplied by 4 (the tens digit of the minute in the third position, except for those already burning in the 1st and 2nd positions) and multiplied by 7 (the units digit of the minute in the fourth position, not equal to the digits in the 1st, 2nd, and 3rd positions). This results in 84 combinations, burning for 84 minutes, which is 1 hour and 24 minutes per day. Together, we get 10 hours and 44 minutes.

## Grading Criteria.

The idea of dividing the consideration into cases for the time of day from 00:00 to 19:59 and from 20:00 to 23:59: 2 points.

The idea of counting the number of options by digits in a convenient order of position: 3 points.

The correct calculation after this: 2 points.