1. Identify the letters and vowels

The word GARDEN has 6 distinct letters: $G, A, R, D, E, N$

So the total number of words formed by arranging all letters is $6! = 720$

The vowels are: $A, E$

2. Interpret “vowels in alphabetical order”

Since the only vowels are $A$ and $E$, alphabetical order means: $A \text{ appears before } E$

3. Count arrangements with vowels in alphabetical order

In all permutations of the 6 distinct letters, the relative order of $A$ and $E$ is equally likely to be:

• $A$ before $E$
• $E$ before $A$

Thus, exactly half of all arrangements have vowels in alphabetical order.

So, $\#(\text{vowels in alphabetical order}) = \frac{6!}{2} = \frac{720}{2} = 360$

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

4. Find the required probability

We need the probability that the selected word will NOT have vowels in alphabetical order: $1 - \frac{1}{2} = \frac{1}{2}$

5. Check options

The correct option is: $\boxed{\frac{1}{2}}$ which is Option B.