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All digits different

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A $24$ -hour digital clock displays the time as $6$ digits $hh : mm : ss$ , with leading zeros, for every second from $00 : 00 : 00$ to $23 : 59 : 59$ , which is $86400$ displays. For how many of these displays are all $6$ digits different from each other?

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Hint

Split the hours by how many of their digits are at most $5$ , because the tens digits of minutes and seconds must be at most $5$ .

Solution

The hours with $2$ different digits are $21$ in number: $13$ of them have both digits at most $5$ ( $01$ to $05$ , $10$ , $12$ to $15$ , $20$ , $21$ , $23$ ) and $8$ have one larger digit ( $06$ to $09$ , $16$ to $19$ ).
For an hour with both digits at most $5$ , choosing the tens digit of the minutes ( $4$ ways), the units digit of the minutes ( $3$ small or $4$ large) and then the seconds digits gives $4 \times (3 \times 2 \times 5 + 4 \times 3 \times 5) = 360$ completions.
For an hour with one larger digit the same count is $5 \times (4 \times 3 \times 5 + 3 \times 4 \times 5) = 600$ completions.
The total is $13 \times 360 + 8 \times 600 = 4680 + 4800 = 9480$ .

Answer: $9480$

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