JEE Mathematics 2025 Question with Solution

Given: The word GARDEN has $6$ distinct letters, with vowels $A$ and $E$.

Find: The probability that a randomly selected arrangement does NOT have vowels in alphabetical order.

Total number of arrangements is

$$6! = 720$$

To count arrangements where the vowels are in alphabetical order, choose positions for $A$ and $E$ and then arrange the remaining $4$ consonants:

$$\binom{6}{2} \cdot 4! = 15 \cdot 24 = 360$$

So,

$$P(\text{vowels in alphabetical order}) = \frac{360}{720} = \frac{1}{2}$$

Therefore,

$$P(\text{NOT in alphabetical order}) = 1 - \frac{1}{2} = \frac{1}{2}$$

Hence, the correct option is D.

Given: The vowels are $A$ and $E$ in all arrangements of GARDEN.

Find: The probability that they are NOT in alphabetical order.

For any arrangement, the relative order of $A$ and $E$ is equally likely to be $A$ before $E$ or $E$ before $A$. Since these two possibilities are symmetric,

$$P(A \text{ before } E) = P(E \text{ before } A) = \frac{1}{2}$$

Alphabetical order means $A$ comes before $E$, so not being in alphabetical order means $E$ comes before $A$.

Therefore, the required probability is $\frac{1}{2}$, so the correct option is D.