Permutations & Combinations
Counting
Grade 11
Question:
<p>Two teams are to play a series of five matches between them. A match ends in a win, loss, or draw for a team. A number of people forecast the result of each match and no two people make the same forecast for the series of matches. The smallest group of people in which one person forecasts correctly for all the matches will contain \(n\) people, where \(n\) is</p>
<p>(1) 81</p>
<p>(2) 243</p>
<p>(3) 486</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: Each match has 3 possible outcomes (win, loss, or draw for one team), so the total number of different forecast sequences for 5 matches is 3^5. To guarantee one correct forecast, we need at least this many people.
<p><strong>Step 1:</strong> Identify outcomes per match. Each match has exactly 3 possible outcomes: Win, Loss, or Draw (for a given team's perspective).</p><p><strong>Step 2:</strong> Count total possible forecast sequences. Each person makes one forecast sequence for all 5 matches. The number of different possible sequences = 3 × 3 × 3 × 3 × 3 = 3^5.</p><p><strong>Step 3:</strong> Apply pigeonhole principle. Since no two people make the same forecast, and we have 3^5 = 243 different possible forecasts, we need at least 243 people to guarantee that one person forecasts all 5 matches correctly.</p><p><strong>Step 4:</strong> Calculate 3^5 = 243.</p><p>∴ Answer: n = 243 (Option B)</p>
Correct Answer: B