Question

The total number of times, the digit ${}^{\prime }{3}^{\prime }$ will be written, when the integers having less than $4$ digits are listed, is

• A. $300$
• B. $310$
• C. $302$
• D. $306$

Answer 1

The correct option is A $300$

$\begin{array}{cccc}\text{Number of}& \text{Pattern}& \text{Numbers of}& \text{Number}\\ \text{times 3 used}& & \text{type}& \text{of times 3}\\ & & & \text{appears}\\ 1& --3& 3\times 9\times 9& 1(3\times 9\\ & -3-& & \times 9)\\ & 3--& & \\ 2& -33& 3\times 9& 2(3\times 9)\\ & 33-& & \\ & 3-3& & \\ 3& 333& 1& 3\\ & & \text{Total}& 300\end{array}$
Any place other than $3$ is filled by $9$ ways as ${}^{\prime }{0}^{\prime }$ can appear anywhere which gives all types of numbers like single digit, two digits, etc.

Alternate Solution:
from $000$ to $99$ any digit will appear $20$ times
and if we don't consider the hundreads pleace in a three digit number then all the digits will appear $20$ times in any hundread interval
Now for the case of $300$ to $399$, $3$ will appear $100$ times on the hundreads place
Hence total number of times $3$ will appear in numbers having less than $4$ digits
$=20\times 10+100=300$