<p>To fill 8 vacancies there are 24 candidates of which 6 are from scheduled castes, 5 are from other backward castes. If 25% and \(12\frac{1}{2}\%\) of the vacancies are reserved for the scheduled castes and other backward castes respectively while the rest are open to all, find the number of ways to make the selection of candidates.</p>
Step-by-Step Solution
Key Concept: Break the problem into independent selections for reserved and open categories: SC gets 2 vacancies (25% of 8), OBC gets 1 vacancy (12.5% of 8), and 5 remain open. Calculate each separately, then multiply using the fundamental counting principle.
<p><strong>Step 1: Identify reserved and open vacancies</strong></p><p>Total vacancies = 8</p><p>SC reserved: 25% of 8 = 2 vacancies</p><p>OBC reserved: 12.5% of 8 = 1 vacancy</p><p>Open vacancies: 8 - 2 - 1 = 5 vacancies</p><p><strong>Step 2: Selection for SC reserved positions</strong></p><p>SC candidates available = 6</p><p>Ways to select 2 from 6 SC candidates: C(6,2) = 15</p><p><strong>Step 3: Selection for OBC reserved positions</strong></p><p>OBC candidates available = 5</p><p>Ways to select 1 from 5 OBC candidates: C(5,1) = 5</p><p><strong>Step 4: Selection for open vacancies</strong></p><p>Remaining candidates = 24 - 2(selected SC) - 1(selected OBC) = 21</p><p>Ways to select 5 from 21 candidates: C(21,5) = 20349</p><p><strong>Step 5: Apply fundamental counting principle</strong></p><p>Total ways = C(6,2) × C(5,1) × C(21,5)</p><p>= 15 × 5 × 20349</p><p>= 75 × 20349</p><p>∴ Answer: <strong>1,526,175</strong></p>
Correct Answer: 1