Question

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

• A. $296$
• B. $300$
• C. $271$
• D. $302$

Answer 1

Answer: (B)

Since $3$ does not occur in $1000$, 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 $xyz$ where $0\le x,yz\le 9$. Let us first count the numbers in which $3$ occurs exactly once.
Since $3$ can occur at one place in ${}^3{C}_1$ ways, there are $\left({}^3{P}_1\right)\left(9\times 9\right)=3\times 9^2$ such numbers. Next, 3 can occur in exactly two places in $\left({}^3{C}_1\right)\left(9\right)=3\times 9$ such numbers. Lastly, $3$ can occur in all three digits in one number only. Hence, the number of tithes $3$ occurs is
$1\times {\left(3\times 9\right)}^2+2\times \left(3\times 9\right)+3\times 1$
$=243+54+3=243+57$
$=300$