Permutations & Combinations
Selection with restrictions
Grade None
Question:
<p>A candidate is required to answer 6 out of 10 questions, which are divided into two groups, each containing 5 questions. He is not permitted to attempt more than 4 questions from either group. The number of different ways in which the candidate can choose 6 questions is</p>
<p>50</p>
<p>150</p>
<p>200</p>
<p>250</p>
Step-by-Step Solution
Key Concept: Use complementary counting: subtract invalid cases (all 5 from one group) from total ways to choose 6 from 10. The constraint 'not more than 4 from either group' means we must exclude selections with 5 from Group A or 5 from Group B.
<p><strong>Step 1: Find total ways without restriction</strong></p><p>Total ways to choose 6 from 10 questions = C(10,6) = 210</p><p><strong>Step 2: Identify invalid cases</strong></p><p>The constraint is: at most 4 from either group. Invalid cases occur when one group contributes 5 questions.</p><p><strong>Step 3: Count invalid cases</strong></p><p>Case 1: All 5 from Group A, 1 from Group B = C(5,5) × C(5,1) = 1 × 5 = 5</p><p>Case 2: All 5 from Group B, 1 from Group A = C(5,5) × C(5,1) = 1 × 5 = 5</p><p>Total invalid cases = 5 + 5 = 10</p><p><strong>Step 4: Apply complementary counting</strong></p><p>Valid ways = Total ways - Invalid ways = 210 - 10 = 200</p><p>∴ Answer: C (200)</p>
Correct Answer: C