Question

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

• A. 250
• B. 300
• C. 350
• D. 400

Answer 1

Answer: (B)

300

We have to focus on numbers less than $1000$ which can be formed by filling $3$ empty block with $10$ possible digits, repetition being allowed. We take $3$ cases
1. When five is at the hundreds place, then there are $10\times 10$ possible cases since the two other spaces can be filled with ten possible natural numbers each.
2. When five is at the tens place, again there are $10\times 10$ possible cases since the first and the last place can be filled by ten possible numbers each time.
3. When five is at the units place which again gives us $10\times 10$ possibilities for the same reason.
Now we must note that for numbers like $555,550,505$ we are supposed to count $3,2,2$ respectively since $5$ appears more than once in these numbers and the good news is that our method takes care of that automatically since a number like $505$ will come twice, once when five is at the hundreds place and then again when five is at the units place. Same thing happens with all such numbers so the final answer is $100+100+100=300.$