<p>In how many ways can first and second prizes in Mathematics, first and second prizes in Physics, first prize in Chemistry and first prize in English be given away to a class of 30 students?</p>
Step-by-Step Solution
Key Concept: Each prize is distinct and independent, so we multiply the number of choices for each position sequentially. Math 1st uses 30 choices, Math 2nd uses 29 remaining, Physics 1st uses 30 again (students can win multiple subject prizes), and so on.
<p><strong>Step 1:</strong> Identify independent prize categories - Math, Physics, Chemistry, and English are separate subjects, so the same student can win prizes in multiple subjects.</p><p><strong>Step 2:</strong> Count choices for each prize sequentially:</p><ul><li>Math 1st prize: 30 choices</li><li>Math 2nd prize: 29 choices (different from 1st)</li><li>Physics 1st prize: 30 choices (independent category)</li><li>Physics 2nd prize: 29 choices (different from Physics 1st)</li><li>Chemistry 1st prize: 30 choices</li><li>English 1st prize: 30 choices</li></ul><p><strong>Step 3:</strong> Apply multiplication principle - total ways = 30 × 29 × 30 × 29 × 30 × 30 = (30 × 29) × (30 × 29) × 30 × 30</p><p>∴ Answer: <strong>(30 × 29) × (30 × 29) × 30 × 30</strong></p>
Correct Answer: (30 × 29) × (30 × 29) × 30 × 30