Animated Solution for Mathematics - Probability: Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

Imagine you are standing in front of a blank board, and you are given the word GARDEN. Your task is to rearrange these six distinct letters to form new words.

You might be tempted to immediately start calculating the total number of permutations, which is $6!=720$. While that is a perfectly valid starting point, I want you to pause. In JEE Advanced, the fastest path is often the one that relies on conceptual insight rather than brute-force calculation.

Many students will dive headfirst into the calculation, trying to count how many words have $A$ before $E$ and how many have $E$ before $A$. They might try to fix the positions of $A$ and $E$ and then arrange the remaining four letters.

While this works, it is prone to calculation errors and wastes precious time. We need a more elegant approach.

Let us look at the vowels: $A$ and $E$. In any word formed by these six letters, these two vowels must occupy two distinct positions.

Because they are distinct, there are only two possible relative orderings: either $A$ appears before $E$, or $E$ appears before $A$. Now, ask yourself: is there any reason for $A$ to prefer appearing before $E$ over $E$ appearing before $A$?

No. The letters are shuffled randomly. Therefore, by the principle of symmetry, the number of arrangements where $A$ precedes $E$ must be exactly equal to the number of arrangements where $E$ precedes $A$.

If we let $N$ be the total number of arrangements, $N_{AE}$ be the number of arrangements where $A$ is before $E$, and $N_{EA}$ be the number of arrangements where $E$ is before $A$, we know that:

Because of symmetry, $N_{AE}=N_{EA}$. This implies that:

The question asks for the probability that the vowels are NOT in alphabetical order. Alphabetical order means $A$ comes before $E$. Therefore, NOT in alphabetical order means $E$ comes before $A$.

The probability is simply:

This is the beauty of combinatorics. By identifying the symmetry, we reduced a potentially tedious counting problem into a simple conceptual realization.

Always look for these shortcuts; they are the hallmark of a true problem-solver. You have successfully navigated the GARDEN, and the answer is $\frac{1}{2}$.