Probability
Classical Probability
Grade 12
Question:
<p>Find the probability that the birthdays of six different persons will fall in exactly two calendar months.</p>
<p>\(\dfrac{341}{12^5}\)</p>
<p>\(\dfrac{341}{12^4}\)</p>
<p>\(\dfrac{341}{12^6}\)</p>
<p>\(\dfrac{341}{12^3}\)</p>
Step-by-Step Solution
Key Concept: Use inclusion-exclusion principle: count ways to distribute 6 persons into exactly 2 months, then divide by total possible arrangements. The constraint 'exactly 2' means both months must be non-empty.
<p><strong>Step 1:</strong> Total ways to assign 6 birthdays to 12 months = 12<sup>6</sup></p><p><strong>Step 2:</strong> Ways to assign all 6 birthdays to at most 2 specific months = 2<sup>6</sup>. Number of ways to choose 2 months = C(12,2) = 66. So total ways in at most 2 months = 66 × 2<sup>6</sup> = 66 × 64 = 4224</p><p><strong>Step 3:</strong> Ways to assign all 6 birthdays to exactly 1 specific month = 1. Number of ways to choose 1 month = 12. So total ways in exactly 1 month = 12 × 1 = 12</p><p><strong>Step 4:</strong> Ways for exactly 2 months = 4224 - 12 = 4212</p><p><strong>Step 5:</strong> Probability = 4212/12<sup>6</sup> = 4212/2985984 = 1053/746496 = <strong>351/248832</strong></p><p>∴ Answer: A</p>
Correct Answer: A