<p>How many words can be made with the letters of the word BHARATI so that all the vowels are consecutive?</p>
Step-by-Step Solution
Key Concept: Treat all consecutive vowels as a single unit, then count permutations of the resulting objects accounting for repeated letters within the vowel block.
<p><strong>Step 1:</strong> Identify the letters in BHARATI: B, H, A, R, A, T, I (total 7 letters with A repeating twice)</p><p><strong>Step 2:</strong> Identify vowels: A, A, I (three vowels, with A repeated)</p><p><strong>Step 3:</strong> Treat all vowels as a single block (AAI). Now we have units: [AAI], B, H, R, T (5 objects with no repetition among objects)</p><p><strong>Step 4:</strong> Arrange these 5 units: 5! = 120 ways</p><p><strong>Step 5:</strong> Within the vowel block [AAI], arrange the vowels accounting for repetition of A: 3!/2! = 3 ways (AAI, AIA, IAA)</p><p><strong>Step 6:</strong> Total arrangements = 5! × (3!/2!) = 120 × 3 = 360</p><p>∴ Answer: <strong>360</strong></p>
Correct Answer: 360