Animated Solution for Mathematics - Permutations and Combinations: How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order

Welcome, future engineers! Today, we are going to dive into a classic combinatorics problem that often appears in the JEE Advanced syllabus. It is not just about finding a number; it is about understanding the hidden symmetry in how we arrange objects.

Let us look at the word GARDEN. It is a six-letter word where all its letters—$G$, $A$, $R$, $D$, $E$, and $N$—are distinct. This is our starting point.

Imagine you have $6$ empty slots on your desk and $6$ distinct tiles. If we were to place them randomly, the total number of permutations is simply $6!$.

Let us calculate that:

This $720$ represents every possible universe where these letters can exist. However, our problem imposes a specific constraint.

The problem asks us to arrange these letters such that the vowels, $A$ and $E$, are in alphabetical order. This means that in any valid arrangement, $A$ must appear to the left of $E$. They do not need to be neighbors; they just need to respect their relative sequence.

In the total $720$ arrangements, there are only two ways these two vowels can be arranged relative to each other: either $A$ comes before $E$, or $E$ comes before $A$. Because the letters are distinct and the positions are symmetric, exactly half of the $720$ arrangements will have $A$ before $E$.

To find our answer, we divide the total arrangements by the number of ways the vowels can be arranged, which is $2!$.

If you ever feel unsure about the symmetry argument, there is a more mechanical, "bulletproof" way to solve this. Let us use the Slot Selection method.

First, we choose $2$ slots out of $6$ to place our vowels. The number of ways to do this is given by the combination formula:

Once we have chosen those two slots, we must place $A$ and $E$ in them. Since the order is fixed ($A$ must be before $E$), there is only $1$ way to place them.

Finally, we have $4$ slots left for our $4$ consonants ($G$, $R$, $D$, $N$). We can arrange these $4$ consonants in $4!$ ways:

Multiplying these together, we get:

Whether you use the symmetry argument or the slot selection method, the result is the same: 360. This is the power of combinatorics—multiple paths leading to the same elegant truth.

Remember, whenever you face a problem with a "fixed relative order" constraint, you can always use the general formula $\frac{n!}{r!}$, where $n$ is the total number of items and $r$ is the number of items with the constraint. Keep practicing, stay curious, and keep falling in love with the logic behind the math!