<p><strong>Match the Column:</strong></p><p>The number of 10 letter permutations comprising 4a's, 3b's and 3c's such that:</p><p><strong>Column-I:</strong></p><p>(A) a's are separated and all b's are together is equal to</p><p>(B) a's are separated and exactly two b's are together is equal to</p><p>(C) no two adjacent letters are identical is equal to</p><p>(D) no two b's are together, no two c's are together is equal to</p><p><strong>Column-II:</strong></p><p>(P) 18</p><p>(Q) 20</p><p>(R) 150</p><p>(S) 180</p>
Step-by-Step Solution
Key Concept: Apply gap method for separation of identical items, treat groups of items as single units, and count arrangements with multiple constraints systematically.
<p><strong>Case A:</strong> a's separated (4 a's with gaps), all b's together (bbb as one unit). Arrange 4 a's creating 5 gaps, place bbb in one gap, arrange 3c's: gives 180 → (S)</p><p><strong>Case B:</strong> a's separated, exactly two b's together. Arrange 4 a's in 5 gaps, place (bb) in one gap, single b and 3c's arranged: gives 20 → (Q)</p><p><strong>Case C:</strong> No two identical adjacent. Complex constraint with 4a's, 3b's, 3c's: gives 150 → (R)</p><p><strong>Case D:</strong> No two b's adjacent, no two c's adjacent. Using alternating arrangement principles: gives 18 → (P)</p>
Correct Answer: A→S, B→Q, C→R, D→P