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 ? x , y , z ? 9 .

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

Since 8 can occur atone place in  3 C 1ways. There are  3 x 9 x 9 such numbers.

Next, 8 can occur in exactly two places in  3 C 2 × 9 such numbers. Lastly, 8 can occur in all three digits in one number only.

Hence, the number of times 8 occur is 

  1 × 3 × 9 2 + 2 × 3 × 9 + 3 × 1 = 300