<p>If the papers of 4 students can be checked by any one of the 7 teachers, then the probability that all the 4 papers are checked by exactly 2 teachers is</p>
Step-by-Step Solution
Key Concept: Use the inclusion-exclusion principle to count assignments where exactly 2 teachers check all 4 papers: choose 2 teachers, subtract cases where only 1 teacher checks all papers, then divide by total assignments 7^4.
<p><strong>Step 1:</strong> Total ways to assign 4 papers to 7 teachers = 7^4 = 2401</p><p><strong>Step 2:</strong> For exactly 2 teachers to check all 4 papers:</p><p>• Choose 2 teachers from 7: C(7,2) = 21</p><p>• Assign 4 papers to these 2 teachers such that both are used</p><p><strong>Step 3:</strong> Ways to assign 4 papers to 2 specific teachers = 2^4 = 16</p><p>• But subtract cases where only 1 teacher checks all papers = 2 (all to teacher 1 OR all to teacher 2)</p><p>• Valid assignments using exactly both teachers = 16 - 2 = 14</p><p><strong>Step 4:</strong> Total favorable outcomes = 21 × 14 = 294</p><p><strong>Step 5:</strong> Probability = 294/2401 = 294/2401 = 42/343 = 6/49</p><p>∴ Answer: D</p>
Correct Answer: D