Question

The number of times the digits $3$ will be written when listing the integers from $1$ to $1000$ is?

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

Answer 1

The correct option is B

$300$

Explanation for the correct option:

Calculate the number of times the given digit occurs.

$3$ does not occur in $1000$. So, we have to count the numbers from $1$ to $999$.
The number from $1$ to $999$ is of the form $xyz$
$0\le x,y,z\le 9$
Number of times in which $3$ occurs exactly once
$={}^3\mathrm{C}_1\times 9\times 9$
Number of times in which $3$ occurs exactly twice
$={}^3\mathrm{C}_2\times 9$
Number of times in which $3$ occurs exactly thrice
$={}^3\mathrm{C}_3\times 1$
Number of times $3$ occurs
$=\left({}^3\mathrm{C}_1\times 9\times 9\right)+\left(2\times {}^3\mathrm{C}_2\times 9\right)+\left(3\times {}^3\mathrm{C}_3\right)$
$$\begin{gathered} =243+54+3 \\ =300 \end{gathered}$$

Hence, Option $\left(\mathrm{B}\right)$ is the correct answer.