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A digital clock uses two digits to display hours, two digits to display minutes and two digits to display seconds, e.g. $10$:$23$:$42$.

How many times between $10$:$00$:$00$ and $11$:$00$:$00$ on the same morning are all six digits different?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

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This problem is taken from the UKMT Mathematical Challenges.