<p>Number of 7 letter dull words in which no two vowels are together, is</p>
Step-by-Step Solution
Key Concept: Apply the constraint that no two vowels are adjacent by arranging consonants first, then inserting vowels in the gaps created.
<p><strong>Solution:</strong> In a dull word, all letters of 'AGAIN' are used but consonants (G, N) can be repeated.</p><p>For 7 letters with no two vowels together:</p><p>First arrange consonants (G, N) with repetition allowed. For 4 consonants in a 7-letter word with 3 vowels that cannot be adjacent:</p><p>Required number of ways $= \binom{5}{3} \times \frac{3!}{2!} \times \left(\frac{4!}{3!} + \frac{4!}{3!} + \frac{4!}{2!2!}\right)$</p><p>$= 30(4 + 4 + 6) = 420$</p><p>∴ Answer is (b) 420</p>
Correct Answer: b