Different digital clock
At how many times between 10 and 11 o'clock are all six digits on a digital clock different?
Age
14 to 16
Challenge level
Problem
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.
Student Solutions
All six digits are different $360$ times between $10$:$00$:$00$ and
$11$:$00$:$00$.
To satisfy the stated condition, the display will have the form $10$:$m_{1}m_{2}$:$s_{1}s_{2}$.
The values of both $m_{1}$ and $s_{1}$ have to be chosen from $2$, $3$, $4$, $5$. So there are four ways of choosing $m_{1}$ and three choices for $s_{1}$. Since four digits have been chose, $m_{2}$ and $s_{2}$ are selected from the remaining six.
Thus the total number of times is $4 \times 3 \times 6 \times 5 = 360$.
To satisfy the stated condition, the display will have the form $10$:$m_{1}m_{2}$:$s_{1}s_{2}$.
The values of both $m_{1}$ and $s_{1}$ have to be chosen from $2$, $3$, $4$, $5$. So there are four ways of choosing $m_{1}$ and three choices for $s_{1}$. Since four digits have been chose, $m_{2}$ and $s_{2}$ are selected from the remaining six.
Thus the total number of times is $4 \times 3 \times 6 \times 5 = 360$.