<p>In a group of 13 cricket players, 4 are bowlers. Then in how many ways can they form a cricket team of 11 players in which at least 2 bowlers are included?</p>
Step-by-Step Solution
Key Concept: Use complementary counting: Total ways to select 11 from 13 minus ways with 0 or 1 bowler. Alternatively, count directly by cases: exactly 2, 3, or 4 bowlers included.
<p><strong>Step 1: Identify constraints</strong></p><p>Total players: 13 (4 bowlers, 9 non-bowlers)<br>Team size: 11 players<br>Condition: At least 2 bowlers required</p><p><strong>Step 2: Use complementary counting</strong></p><p>Total ways to select 11 from 13 = C(13,11) = C(13,2) = 78</p><p><strong>Step 3: Subtract unfavorable cases</strong></p><p>Ways with 0 bowlers: C(4,0) × C(9,11) = 1 × 0 = 0 (impossible, only 9 non-bowlers)<br>Ways with 1 bowler: C(4,1) × C(9,10) = 4 × 0 = 0 (impossible, only 9 non-bowlers)</p><p><strong>Step 4: Calculate favorable outcomes</strong></p><p>Ways with at least 2 bowlers = 78 - 0 - 0 = 78</p><p><strong>Verification by direct counting:</strong></p><p>Exactly 2 bowlers: C(4,2) × C(9,9) = 6 × 1 = 6<br>Exactly 3 bowlers: C(4,3) × C(9,8) = 4 × 9 = 36<br>Exactly 4 bowlers: C(4,4) × C(9,7) = 1 × 36 = 36<br>Total = 6 + 36 + 36 = 78 ✓</p><p><strong>∴ Answer: 78 (Option D)</strong></p>
Correct Answer: D