Animated Solution for Mathematics - Permutations and Combinations: The sum of all the 4-digit distinct numbers that can be formed with the digits 1,2,2 and 3 is:

Solving the 4-Digit Puzzle

Welcome, future engineers! Today, we are going to dismantle a classic JEE-style combinatorics problem. It is easy to look at a question like 'Find the sum of all 4-digit numbers formed by 1, 2, 2, and 3' and immediately reach for a pen to start listing numbers.

But stop! If you start listing, you are already falling into the trap of brute force. In the JEE Advanced, time is your most precious currency. We don't want to list; we want to understand the structure.

The Challenge of Repetition

We are given the set $\{1,2,2,3\}$. The immediate 'catch' here is the repetition of the digit $2$.

If all digits were distinct, we would have $4!=24$ numbers. But because the $2$ repeats, we must use the formula for permutations of a multiset:

Knowing there are $12$ numbers is our first victory. Now, how do we sum them without writing them all down?

The Power of Columnar Symmetry

Imagine these $12$ numbers stacked in a column. Instead of looking at the numbers horizontally, look at them vertically. We want to find the sum of all digits in the units place, then the tens place, and so on. This is the 'Sum by Place Value' strategy.

Let's focus on the units place. If we fix the digit $1$ at the units place, we are left with the digits $\{2,2,3\}$ to fill the remaining three positions. The number of ways to arrange these is:

So, the digit $1$ appears $3$ times in the units column.

Now, what if we fix the digit $2$ at the units place? We are left with $\{1,2,3\}$. These are all distinct! The number of ways to arrange them is $3!=6$. So, the digit $2$ appears $6$ times in the units column.

Finally, fix the digit $3$ at the units place. We are left with $\{1,2,2\}$. Again, we have a repeated $2$, so the number of arrangements is:

The digit $3$ appears $3$ times.

The Elegant Calculation

Now, let's sum the digits in the units column:

Here is the beautiful part: because the set of digits is the same regardless of which position we look at (Thousands, Hundreds, Tens, or Units), the sum of the digits in every column must be $24$. This is the power of symmetry. We don't need to repeat the calculation for the other columns; we have already unlocked the secret.

Now, we assemble our result. The total sum is the sum of the digits in each place multiplied by their respective place value:

We can factor out the $24$ to make the arithmetic trivial:

Performing the final multiplication:

There you have it! We didn't list a single number, yet we found the sum of all $12$ of them with absolute precision. This is the mindset of a topper: look for the symmetry, exploit the structure, and let the math do the heavy lifting for you.