# Different Digital Clock

At how many times between 10 and 11 o'clock are all six digits on a digital clock different?

## 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 $4 \times 3 \times 6 \times 5 = 360$.