<p>Seven athletes are participating in a race. In how many ways can the first three athletes win the prizes?</p>
Step-by-Step Solution
Key Concept: This is a permutation problem where order matters (1st, 2nd, 3rd prizes are distinct). We select 3 athletes from 7 where positions are ordered, so we use P(7,3) = 7!/(7-3)!.
<p><strong>Step 1:</strong> Identify the problem type. We need to award three distinct prizes (1st, 2nd, 3rd) to 7 athletes, so order matters.</p><p><strong>Step 2:</strong> Use the permutation formula P(n,r) = n!/(n-r)!</p><p><strong>Step 3:</strong> Calculate P(7,3) = 7!/(7-3)! = 7!/4! = (7 × 6 × 5 × 4!)/(4!) = 7 × 6 × 5</p><p><strong>Step 4:</strong> Compute: 7 × 6 × 5 = 42 × 5 = 210</p><p><strong>Alternative approach:</strong> First prize: 7 choices, Second prize: 6 remaining choices, Third prize: 5 remaining choices. Total = 7 × 6 × 5 = 210</p><p>∴ Answer: <strong>210</strong></p>
Correct Answer: 210