Probability
Distribution Problems
Grade 12
Question:
<p>If the papers of 4 students can be checked by any one of the 7 teachers. If the probability that all the 4 papers are checked by exactly 2 teachers is A, then the value of \(490A\) must be ……….</p>
Step-by-Step Solution
Key Concept: Use inclusion-exclusion to count distributions of papers to exactly 2 teachers, ensuring both teachers check at least one paper.
<p><strong>Solution:</strong></p><p>Each of 4 papers can be checked by any of 7 teachers.</p><p>Total ways = \(7^4 = 2401\)</p><p>Favorable outcomes: exactly 2 teachers check all 4 papers</p><p>Step 1: Choose 2 teachers from 7: \(\binom{7}{2} = 21\)</p><p>Step 2: Distribute 4 papers among 2 chosen teachers such that both check at least one paper</p><p>Ways = \(2^4 - 2 = 16 - 2 = 14\) (total distributions minus the cases where only one teacher checks all)</p><p>Favorable outcomes = \(21 \times 14 = 294\)</p><p>Probability A = \(\frac{294}{2401} = \frac{294}{2401}\)</p><p>\(490A = 490 \times \frac{294}{2401} = \frac{490 \times 294}{2401} = \frac{144060}{2401} = 60\)</p>
Correct Answer: 60