<p><strong>254.</strong> The number of words which can be formed using all the letters of the word "NARENDRABHAI" such that no two non-repeated letters occur together is:</p>
Step-by-Step Solution
Key Concept: First identify repeated letters (A appears 3 times, R appears 2 times), then use the restriction that non-repeated letters (N, E, D, B, H, I) must be separated by repeated letters. Arrange repeated letters first, then place non-repeated letters in the gaps created.
<p><strong>Step 1:</strong> Identify the letters in 'NARENDRABHAI'</p><p>Repeated: A (3 times), R (2 times)</p><p>Non-repeated: N, E, D, B, H, I (6 letters, all distinct)</p><p><strong>Step 2:</strong> Arrange the 5 repeated letters (A, A, A, R, R)</p><p>Number of arrangements = 5!/(3!×2!) = 120/12 = 10</p><p><strong>Step 3:</strong> These 5 letters create 6 gaps: _A_A_A_R_R_</p><p><strong>Step 4:</strong> We have exactly 6 non-repeated letters and 6 gaps. To ensure no two non-repeated letters are together, place exactly one non-repeated letter in each gap.</p><p>Number of ways = 6! = 720</p><p><strong>Step 5:</strong> Total arrangements = 10 × 720 = 7200</p><p>∴ Answer: C</p>
Correct Answer: C