Animated Solution for Mathematics - Permutations and Combinations: The number of times the digit 3 will be written when listing the integers from 1 to 1000 is

Welcome, future engineer! Today, we are not just solving a counting problem; we are embarking on a journey to understand the architecture of numbers.

Imagine you are standing before a massive library, and your task is to count every single instance of the digit $3$ in the pages of a book containing numbers from $1$ to $1000$. It sounds daunting, but as we peel back the layers, you will see the elegance hidden in the chaos.

The Range Transformation

First, let's address the elephant in the room: the number $1000$. Does it contain the digit $3$? Absolutely not. So, we can safely set it aside.

Now, consider the range from $000$ to $999$. By padding our numbers with leading zeros, we transform every integer into a $3$-digit sequence.

This is the power of perspective! We have $1000$ numbers, each with $3$ slots: $[H][T][U]$. This transformation turns a messy counting problem into a structured combinatorial one.

The Slot Method

Let's isolate the digit $3$. If we fix a $3$ in the units place, we have $10$ choices for the hundreds place ($0-9$) and $10$ choices for the tens place ($0-9$).

That gives us $10\times 10=100$ occurrences. By symmetry, the same logic applies to the tens place and the hundreds place.

Each position hosts the digit $3$ exactly $100$ times. When we sum these up, we get:

The Symmetry Shortcut

Here is the masterstroke. In our range of $000$ to $999$, we have $1000$ numbers, each with $3$ digits. That is $3000$ total digits written.

Since every digit from $0$ to $9$ is perfectly symmetric, each digit must appear exactly:

It is beautiful, isn't it? The math aligns perfectly, and the logic is undeniable. The total number of times the digit $3$ appears is $300$.

Keep pushing forward. Every problem you solve is a brick in the foundation of your engineering career. You have the tools; now go out and build something incredible!