The number of times the digit 3 will be written when listing the integers from 1 to 1000, is

Options

A. 300
B. 297
C. 243
D. 273

Explanation

Since 3 does not occur in 1000. So, we have to count the number of times 3 occurs when we list the integers from 1 to 999. Any number between 1 and 999 is of the form abc where `0lea,b,cle9`. Clearly,
Number of times 3 occurs
=(No. of number in which 3 occurs exactly at one place) +2 (No. of numbers in which 3 occurs exactly at two places) +3(No. of numbers in which 3 occurs exactly at three places)
Now
No. of numbers in which 3 occurs exactly at one place:
Since 3 can occur at one place in `^(3)C_(1)` ways and each of the remaining two places can be filled in 9 ways. So, number of numbers in which 3 occurs exactly at one place `=^(3)C_(1)xx9xx9`.
No. of number in which 3 occurs exactly at two places : Since 3 can occru exactly at two places in `^(3)C_(2)` ways and the remaining place can be filled in 9 ways.
So, number of numbers in which 3 occurs exactly at two places
`=""^(3)C_(2)xx9`.
No. of numbers in which 3 occurs at all the three places:
Since 3 can occur in all the three digits in one way only. So, number of numbers in which 3 occurs at all the three places is one.
Hence, Numbers of times 3 occurs
`=""^(3)C_(1)xx9xx9+2(""^(3)C_(2)xx9)+3xx1=300`.