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

Options

A. 269
B. 300
C. 271
D. 302

Explanation

Since 3 does not occur in1000. 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 from abc, where `0lea,b,cle0`
Clearly, Number of times 3 occurs
= (No. of numbers in which 3 occrus exactly at one place)
+ 2 (No. of numbers is 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 numbers in which 3 occurs exactly at two places:
Since 3 can occur 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 all the three places is one.
Hence, Numbers of times 3 occurs
`=""^(3)C_(1)xx9xx+2(""^(3)C_(2)xx9)+3xx1=300`