How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?

The word GARDEN has 6 distinct letters:

Vowels: A, E (2 letters)

Consonants: G, R, D, N (4 letters)

Calculate Total Arrangements

Without restrictions, the total number of ways to arrange 6 unique letters is 6! (6-factorial):

$$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$$

Apply the Vowel Restriction

For any random arrangement, the two vowels (A and E) can only appear in $$2! = 2$$ relative orders:

1. A appears before E (Alphabetical order: A...E)

2. E appears before A (Reverse order: E...A)

Exactly half of the total arrangements will feature the vowels in alphabetical order so we divide the total arrangements by 2:

$$\frac{720}{2} = 360$$

Therefore, there are 360 ways to arrange the letters in the word GARDEN so that the vowels appear in alphabetical order.