Permutations & Combinations
Selection with no two adjacent
Grade 11
Question:
<p>If \(n\) objects are arranged in a row, then the number of ways of selecting three of these objects so that no two of them are next to each other is</p>
<p>\({}^{n-2}C_3\)</p>
<p>\({}^{n-3}C_2\)</p>
<p>\({}^{n-3}C_3\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: When selecting 3 non-adjacent objects from n arranged objects, use the gap method: arrange the remaining (n-3) objects first to create (n-2) valid positions for placing the 3 selected objects.
<p><strong>Step 1:</strong> Understand the constraint. We need 3 objects from n objects such that no two selected objects are consecutive.</p><p><strong>Step 2:</strong> Use the gap method. First, arrange the (n-3) unselected objects in a row. These create (n-3)+1 = (n-2) gaps (including ends) where we can place selected objects.</p><p><strong>Step 3:</strong> We need to choose 3 gaps from these (n-2) available gaps to place our 3 selected objects. This ensures no two selected objects are adjacent.</p><p><strong>Step 4:</strong> The number of ways = C(n-2, 3) = (n-2)!/(3!(n-5)!) = (n-2)(n-3)(n-4)/6</p><p><strong>Note:</strong> This formula is valid for n ≥ 5.</p><p>∴ Answer: <strong>C(n-2, 3)</strong></p>
Correct Answer: A