Question

The number of times the digit $5$ will be written while listing the intergers from $1$ to $1000$ is

• A. $289$
• B. $299$
• C. $300$
• D. None of these

Answer 1

Answer: (C)

$300$

Any number 1 and 999 is of the form $xyz$ where $0\le x,y,x\le 9$

Let us first count the numbers in which 3 occurs exactly once. Since 3 can occur at one place in ${{}^{3}C}_1$ ways, thus there are ${{}^{3}C}_1\times 9\times 9=243$ such numbers.

Next 3 can occur in exactly 3 places in ${{}^{3}C}_2\times 9=27$ ways

Lastly, 3 can occur in all the 3 digits in only 1 number (333). So number of times 3 occurs are $243+2\times 27+3=300$