Question

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

Answer 1

Evaluate the number of times the given digit will be listed in between the given set of integers

Given integers, $1$ to $1000$

We know that the numbers having the digit $3$ from $1$ to $100$ are as follows,

$3$ at units digit $\to 3,13,23,33,43,53,63,73,83,93$

$3$ at tens digit $\to 30,31,32,33,34,35,36,37,38,39$

So, the number of times the digit $3$ is written from $1$ to $100=20$ [$\because 33$ has two $3$'s]

The same pattern would be followed in the numbers $101-200,201-300,301-400,401-500,501-600,601-700,701-800,801-900$ and $901-1000$.

Then, the number of times the digit $3$ is written as units and tens digit from $1$ to $1000=20\times 10$

$\Rightarrow$ $=200$

Again, as we know, the digit $3$ is written as hundreds digit from $300$ to $399$.

So, the number of times the digit $3$ is written as hundreds digit from $300$ to $399=100$

Thus, the total number of times the digit $3$ is written from $1$ to $1000=200+100$

$\Rightarrow$ $=300$

Hence, the digit $3$ will be written $300$ times when listing the integers from $1$ to $1000$.