The Number of times the digit 8 will be written when listing the integers from 1 to 1000 is :

 Since 8 does not occur in 1000, we have to count the number of times 8 occurs when we list the integers from 1 to 999. Any number between 1 and 999 is of the form xyz, where  $0\le x,y,z\le 9.$

Let us first count the numbers in which 8 occurs exactly once.

Since 8 can occur atone place in  $3C1$ways. There are  $3 x 9 x 9$ such numbers.

Next, 8 can occur in exactly two places in  $3C2\times 9$ such numbers. Lastly, 8 can occur in all three digits in one number only.

Hence, the number of times 8 occur is 

  $1\times 3\times 92+2\times 3\times 9+3\times 1 = 300$