<p>How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?</p>
Step-by-Step Solution
Key Concept: Fix the relative order of vowels (A, E must appear in that order) while counting all arrangements of 6 letters. Vowels must occupy 2 of the 6 positions, and their order is predetermined, so we only choose positions.
<p><strong>Step 1:</strong> Identify the letters in GARDEN: G, A, R, D, E, N (6 letters total: 4 consonants, 2 vowels)</p><p><strong>Step 2:</strong> The vowels are A and E. We need them in alphabetical order (A before E always).</p><p><strong>Step 3:</strong> Choose 2 positions out of 6 for the vowels: C(6,2) = 15 ways. Since the order is fixed (A before E), there's only 1 way to arrange the vowels once positions are chosen.</p><p><strong>Step 4:</strong> Arrange the 4 consonants (G, R, D, N) in the remaining 4 positions: 4! = 24 ways</p><p><strong>Step 5:</strong> Total arrangements = C(6,2) × 4! = 15 × 24 = 360</p><p>∴ Answer: C (360)</p>
Correct Answer: C