<p>In how many ways can the letters of the word ALGEBRA be arranged without changing the relative order of vowels and consonants?</p>
Step-by-Step Solution
Key Concept: Fix the positions of vowels (A, E, A) and consonants (L, G, B, R) separately, then count arrangements within each group while maintaining their relative order. The vowels and consonants must occupy their fixed 'slots' in the overall arrangement.
<p><strong>Step 1:</strong> Identify vowels and consonants in ALGEBRA:</p><p>Vowels: A, E, A (3 vowels, with A repeated twice)</p><p>Consonants: L, G, B, R (4 consonants, all distinct)</p><p></p><p><strong>Step 2:</strong> Determine positions. In any arrangement of 7 letters, we need to choose 3 positions for vowels and 4 for consonants. Since relative order within each group must be preserved, there is only one way to place them: C(7,3) ways to choose which 3 of 7 positions are for vowels.</p><p></p><p><strong>Step 3:</strong> Count arrangements of vowels in their 3 chosen positions:</p><p>Vowels are A, E, A → arrangements = 3!/2! = 3 (since A repeats twice)</p><p></p><p><strong>Step 4:</strong> Count arrangements of consonants in their 4 chosen positions:</p><p>Consonants are L, G, B, R (all distinct) → arrangements = 4! = 24</p><p></p><p><strong>Step 5:</strong> Calculate total arrangements:</p><p>Total = C(7,3) × (3!/2!) × 4! = 35 × 3 × 24 = 2520</p><p></p><p>∴ <strong>Answer: 2520</strong></p>
Correct Answer: 2520