Question

The number of times the digit 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

To find the number of times 3 occurs in listing the integer from 1 to 999. (since 3 does not occur in 1000). Any number between 1 to 999 is a 3 digit number xyzz where the digit x,y,z are any digits from 0 to 9.

Now, we first count the numbers in which 3 occurs once only. Since 3 can occur at one place in ${}^3{C}_1$ ways, there are ${}^3{C}_1.(9\times 9)$ = ${3.9}^2$ such numbers.

Again, 3 can occur in exactly two places in ${}^3{C}_1\left(9\right)$ such numbers. Lastly 3 can occur in all the three digits in one such number only 333.

$\therefore$The number of times 3 occurs is equal to $1\times (3\times 9^2)+2\times (3\times 9)+3\times 1$=300.