The number of times the digit 5 will be written when listing the...

Any number& between $1$ to $999$ is of the form $abc$ when $0 \le a, b, c \le 9$. Let us first count the number in which $5$ occurs exactly once.

Since, $5$ can occur at one place in $1 \times C1 3 \times 9 \times 9 = 243$ ways, next $5$ can occur in exactly two places in $C2 3 \times 9 = 27$.

Lastly, $5$ can occur in all three digits in only one way.

Hence, the number of times $5$ occurs

$= 1 \times 243 + 27 \times 2 + 1 \times 3$

$= 243 + 54 + 3 = 300$