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(D) The word $	ext{GARDEN}$ contains $6$ distinct letters: $\{G, A, R, D, E, N\}$.Total number of arrangements $= 6! = 720$.The vowels in the word ar

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$\frac{1}{2}$

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(D) The word $	ext{GARDEN}$ contains $6$ distinct letters: $\{G, A, R, D, E, N\}$.Total number of arrangements $= 6! = 720$.The vowels in the word are $\{A, E\}$.In any arrangement,the vowels $A$ and $E$ can appear in two relative orders: $(A, E)$ or $(E, A)$.Since there are only two vowels,these two orders are equally likely.Thus,the probability that the vowels appear in alphabetical order $(A, E)$ is $\frac{1}{2}$.The probability that the vowels do $	ext{NOT}$ appear in alphabetical order is $1 - P(	ext{alphabetical order}) = 1 - \frac{1}{2} = \frac{1}{2}$.

Let $S$ be the set of all the words that can be formed by arranging all the letters of the word $\text{GARDEN}$. From the set $S$,one word is selected at random. The probability that the selected word will $\text{NOT}$ have vowels in alphabetical order is:

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