<p>In how many ways can a team of 11 players be formed out of 25 players, if 6 out of them are always to be included and 5 always to be excluded?</p>
Step-by-Step Solution
Key Concept: After fixing 6 players (always included) and excluding 5 players, you need to select the remaining positions from the available pool. This reduces the problem from C(25,11) to C(14,5).
<p><strong>Step 1:</strong> Identify the constraints.</p><ul><li>Total players = 25</li><li>Players to be included (fixed) = 6</li><li>Players to be excluded (fixed) = 5</li><li>Team size required = 11</li></ul><p><strong>Step 2:</strong> Calculate remaining selections needed.</p><ul><li>Since 6 players are already included, we need 11 - 6 = 5 more players</li></ul><p><strong>Step 3:</strong> Determine the available pool.</p><ul><li>Total players = 25</li><li>Subtract fixed inclusions = 25 - 6 = 19</li><li>Subtract fixed exclusions = 19 - 5 = 14</li><li>Pool from which we select = 14 players</li></ul><p><strong>Step 4:</strong> Calculate combinations.</p><p>We need to select 5 players from the remaining 14 players:</p><p>C(14, 5) = 14!/(5! × 9!) = (14 × 13 × 12 × 11 × 10)/(5 × 4 × 3 × 2 × 1) = 2,002</p><p><strong>∴ Answer: B (2,002)</strong></p>
Correct Answer: B