Animated Solution for Mathematics - Permutations and Combinations: If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to

A Journey into Permutations

Welcome, aspiring mathematician. Today, we are not just solving a problem; we are embarking on a journey to understand the fundamental architecture of numbers.

We are tasked with forming numbers greater than $10,000$ using the set of digits $\{0,2,4,6,8\}$ without repetition. This might seem like a simple counting exercise, but it is a gateway to the beautiful world of combinatorics.

The Geometric Reality of the Number

First, let us orient ourselves. We have five distinct digits: $\{0,2,4,6,8\}$. We need to form numbers strictly greater than $10,000$.

A number greater than $10,000$ must have at least five digits. Since we only have five unique digits available and we are forbidden from repeating any of them, we are forced into a specific geometric reality: every valid number we form must be a $5$-digit number.

We cannot form a $6$-digit number because we would run out of digits, and a $4$-digit number would be inherently smaller than $10,000$. Thus, we are looking to fill exactly five slots: the ten-thousands, thousands, hundreds, tens, and units places.

The Leading Zero Trap

Now, here is where the story gets interesting. We have five slots to fill. Let us look at the first slot—the ten-thousands place.

If we were to place a $0$ here, what happens? A number like $02468$ is mathematically equivalent to $2468$. This is a $4$-digit number, and it is clearly less than $10,000$.

This is the "Leading Zero Trap." To satisfy our condition, the first digit must be non-zero. Looking at our set $\{0,2,4,6,8\}$, the valid candidates for the first slot are $\{2,4,6,8\}$. This gives us exactly $4$ choices for the first position.

The Permutation Logic

With the first slot filled, we turn our attention to the remaining four slots. We started with five digits and used one. That leaves us with exactly four digits remaining.

The restriction on the zero has vanished! Now that the first slot is occupied by a non-zero digit, the zero is free to take its place in any of the thousands, hundreds, tens, or units positions.

We have four empty slots and four distinct digits to place in them. Since the order of these digits creates a unique number, we are looking at a permutation of $4$ items. The number of ways to arrange $4$ distinct items is $4!$, which is calculated as:

The Final Synthesis

We have arrived at the climax of our journey. We have $4$ independent choices for the first digit and $24$ ways to arrange the remaining digits for each of those choices.

According to the fundamental principle of counting, we multiply these possibilities together:

There it is. Ninety-six. It is not just a number; it is the result of logical deduction, careful constraint analysis, and the elegant application of the multiplication principle.

Keep this clarity of thought, and you will find that even the most complex problems in JEE Advanced are just stories waiting to be told.