Permutations & Combinations
Selection with conditions
Grade 11
Question:
<p>At least 4 from the first 5 questions means either 4 or 5 from the first 5 questions, and the rest from the remaining 8 questions. The number of ways to attempt a paper of 13 questions (at least 4 from first 5) choosing exactly 6 questions in total is:</p>
<p>196</p>
<p>140</p>
<p>280</p>
<p>56</p>
Step-by-Step Solution
Key Concept: Break the constraint into mutually exclusive cases: exactly 4 from first 5 (then 2 from remaining 8) OR exactly 5 from first 5 (then 1 from remaining 8). Add these cases using the sum rule of counting.
<p><strong>Step 1:</strong> Interpret the constraint. 'At least 4 from first 5' while choosing 6 total means either:</p><ul><li>Case 1: Exactly 4 from first 5 questions + exactly 2 from remaining 8</li><li>Case 2: Exactly 5 from first 5 questions + exactly 1 from remaining 8</li></ul><p><strong>Step 2:</strong> Calculate Case 1: C(5,4) × C(8,2) = 5 × 28 = 140</p><p><strong>Step 3:</strong> Calculate Case 2: C(5,5) × C(8,1) = 1 × 8 = 8</p><p><strong>Step 4:</strong> Apply addition rule (cases are mutually exclusive):</p><p>Total = 140 + 8 = <strong>148</strong></p><p>∴ Answer: A</p>
Correct Answer: A