<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, N, R, D, B, H, I appear once each). The constraint 'no two non-repeated letters occur together' means the 6 distinct single letters cannot be adjacent to each other—they must be separated by the repeated A's.
<p><strong>Step 1:</strong> Identify letter frequencies in "NARENDRABHAI":<br/>A appears 3 times; N, R, E, D, B, H, I appear 1 time each (6 distinct single letters)</p><p><strong>Step 2:</strong> Interpret constraint: The 6 non-repeated letters cannot be adjacent. This means they must be separated by the 3 A's.</p><p><strong>Step 3:</strong> Arrange the 3 A's in a line: A_A_A_<br/>This creates exactly 4 gaps (before first A, between A's, after last A) where the 6 distinct letters must be placed.</p><p><strong>Step 4:</strong> We need to place 6 distinct letters in 4 gaps such that no gap has more than 1 letter (otherwise two non-repeated letters would be adjacent).<br/>This is impossible! We have 6 letters but only 4 gaps.</p><p><strong>Step 5:</strong> Reinterpret: Arrange 3 A's and 6 distinct letters such that no two of the 6 distinct letters are adjacent.<br/>First arrange 3 A's: 1 way<br/>This creates 4 positions (gaps): _A_A_A_<br/>Choose 4 of the 6 distinct letters to place in these gaps: C(6,4)<br/>Arrange these 4 letters in gaps: 4!<br/>Remaining 2 letters must go within the A positions (impossible without violating constraint)</p><p><strong>Step 6:</strong> Correct approach: Arrange 6 distinct letters first: 6! ways<br/>This creates 7 gaps (before, after, and between each letter): _L_L_L_L_L_L_<br/>Place 3 identical A's in these 7 gaps to ensure no two L's are adjacent: C(7,3)<br/>Total arrangements = 6! × C(7,3) = 720 × 35 = 25,200</p><p>∴ Answer: C</p>
Correct Answer: C