The number of times the digit $5$ will be written when listing the integers from $1$ to $1000$, is

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 $1\times {}^{3}C_1\times 9\times 9=243$ ways, next $5$ can occur in exactly two places in ${}^3\mathrm{C}_2\times 9=27$.

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

Hence, the number of times $5$ occurs

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

$=243+54+3=300$