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Answer:

272 times.

 

Logic:

Assuming a standard 12-hour digital clock that does not use a leading zero for single-digit hours (e.g., it shows 8:00, not 08:00), we need to count the valid combinations for the tens-of-minutes (0-5) and the single-minutes (0-9) that do not repeat any digit already on the screen.


Here is the breakdown by hour:

  • The 8:00 & 9:00 hours: * The hour takes up one digit (8 or 9).

    • The tens-minute can be anything from 0 to 5 (6 options).

    • The single-minute can be any of the 10 digits minus the hour digit and the tens-minute digit (8 options).

    • 6 x 8 = 48 times (for the 8:00 hour)

    • 6 x 8 = 48 times (for the 9:00 hour)

  • The 10:00 hour: * The hour uses the digits '1' and '0'.

    • The tens-minute can only be 2, 3, 4, or 5 (4 options).

    • The single-minute has 7 remaining options.

    • 4 x 7 = 28 times

  • The 11:00 hour: * The hour uses '1' and '1'.

    • 0 times (It already contains a repeating digit).

  • The 12:00 hour: * The hour uses the digits '1' and '2'.

    • The tens-minute can be 0, 3, 4, or 5 (4 options).

    • The single minute has 7 remaining options.

    • 4 x 7 = 28 times

  • The 1:00, 2:00, & 3:00 hours: * The hour takes up one digit.

    • The tens-minute has 5 options (0 to 5, excluding the specific hour digit).

    • The single-minute has 8 remaining options.

    • 5 x 8 = 40 times (for the 1:00 hour)

    • 5 x 8 = 40 times (for the 2:00 hour)

    • 5 x 8 = 40 times (for the 3:00 hour)


(Note: 4:00 PM exactly is 4:00, which has repeating zeros, so it does not add to the count.)


Total Calculation: 48 + 48 + 28 + 0 + 28 + 40 + 40 + 40 = 272

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