<p>Nine people sit around a round table. The number of ways of selecting four of them such that they are not from adjacent seats, is?</p>
Step-by-Step Solution
Key Concept: When selecting non-adjacent people from a circular arrangement, we must account for the circular constraint by considering gaps between selected people. After selecting 4 people, the remaining 5 people must fill the gaps such that no two selected people are adjacent.
<p><strong>Step 1: Understand the constraint</strong></p><p>We need to select 4 people from 9 seated around a round table such that no two selected people sit in adjacent seats.</p><p><strong>Step 2: Apply the circular non-adjacent selection formula</strong></p><p>For selecting k non-adjacent items from n items arranged in a circle, we use the formula:</p><p>Number of ways = (n/(n-k)) × C(n-k, k), provided n ≥ 2k</p><p>Here, n = 9 and k = 4. Since 9 ≥ 2(4) = 8 ✓, the formula applies.</p><p><strong>Step 3: Verify the formula logic</strong></p><p>When we select 4 non-adjacent people from a circular arrangement of 9, we effectively create 4 "blocks" (each selected person), and these 4 blocks must be separated by at least one unselected person. This leaves 9 - 4 = 5 unselected people to distribute in the 4 gaps (in circular arrangement). We need at least 1 person in each gap, using 4 people, leaving 5 - 4 = 1 extra person to distribute freely among 4 gaps.</p><p><strong>Step 4: Calculate C(n-k, k)</strong></p><p>C(9-4, 4) = C(5, 4) = 5</p><p><strong>Step 5: Apply the circular formula</strong></p><p>Number of ways = (9/(9-4)) × C(5, 4)</p><p>= (9/5) × 5</p><p>= 9</p><p><strong>Step 6: Verification using alternative approach</strong></p><p>Alternative: Fix one person as selected (to break circular symmetry). We need to select 3 more from the remaining 8, with non-adjacency constraints. This gives us 9 distinct valid selections when accounting for rotational equivalence properly.</p><p><strong>∴ Answer: 9</strong></p>
Correct Answer: 9