Question:

<p><strong>Step 1:</strong> For circular permutations with constraint, fix one person's position. Fix one gentleman (say G₁) to eliminate rotational counting.</p><p><strong>Step 2:</strong> We need exactly 2 ladies adjacent. Treat 2 ladies (L₁, L₂) as a single block and keep 1 lady (L₃) separate (not adjacent to any other lady).</p><p><strong>Step 3:</strong> Arrange around the fixed gentleman: the block of 2 ladies, 2 remaining gentlemen (G₂, G₃), and 1 isolated lady. This gives 4 objects to arrange in (4-1)! = 3! = 6 ways (since G₁ is fixed as reference).</p><p><strong>Step 4:</strong> Within the block of 2 ladies, arrange L₁ and L₂: 2! = 2 ways.</p><p><strong>Step 5:</strong> Arrange the other 3 ladies among themselves (choosing which 2 sit together and which is isolated): C(3,2) = 3 ways to choose which 2 ladies form the block.</p><p><strong>Step 6:</strong> Arrange the 2 gentlemen (other than fixed G₁): 2! = 2 ways.</p><p><strong>Step 7:</strong> Total = 3 × 2! × 2! × 2 = 3 × 2 × 2 × 2 = 24... (recalculating with correct circular logic)</p><p><strong>Correct Approach:</strong> Fix G₁. Arrange block(2L), 2 other gentlemen, and 1L in remaining 5 positions: 4! arrangements. Choose 2 ladies for block: C(3,2) = 3. Arrange 2 ladies within block: 2!. Arrange 2 gentlemen: 2!. Total = 3 × 2! × 2! × 4! = 3 × 2 × 2 × 24 = 288... (requires refinement)</p><p><strong>Final Correct Solution:</strong> Fix 1 gentleman. Remaining 5 people: arrange as [block of 2L] + [3G including those with gaps] ensuring only one pair of L adjacent. Systematic counting: 3(ways to choose 2L for block) × 2!(arrange within block) × 2!(other 2G) × 3!(arrange 4 objects around fixed G with L₃ isolated) = 3 × 2 × 2 × 6 = 72</p><p>∴ <strong>Answer: 72</strong></p>