In a cricket club there are 6 bowlers and 9 batsmen. In how many different ways can a team of 11 players be selected such that the team contains at least 4 bowlers?

Introduction / Context:
This question involves combinations with category based constraints. We need to form an 11 player cricket team from a pool of bowlers and batsmen, ensuring that the team has at least 4 bowlers. It is a classic example of selecting with lower bounds on one category.

Given Data / Assumptions:

• Total bowlers available: 6.
• Total batsmen available: 9.
• Team size must be exactly 11 players.
• The team must contain at least 4 bowlers.
• All players are distinct; order of selection does not matter.

Concept / Approach:
We break the problem into cases based on the exact number of bowlers in the team: 4, 5 or 6 bowlers (at least 4 means 4 or more, but there are only 6 bowlers available). For each case, we choose the corresponding number of bowlers and batsmen using combinations and then sum the counts from all valid cases.

Step-by-Step Solution:
Step 1: Case 1 (4 bowlers): Choose 4 bowlers out of 6 and 7 batsmen out of 9. The number of ways is 6C4 * 9C7.Step 2: Compute 6C4 = 6C2 = 15 and 9C7 = 9C2 = 36. Thus, Case 1 gives 15 * 36 = 540 ways.Step 3: Case 2 (5 bowlers): Choose 5 bowlers out of 6 and 6 batsmen out of 9. The number of ways is 6C5 * 9C6.Step 4: Compute 6C5 = 6C1 = 6 and 9C6 = 9C3 = 84. Thus, Case 2 gives 6 * 84 = 504 ways.Step 5: Case 3 (6 bowlers): Choose all 6 bowlers and then 5 batsmen out of 9. The number of ways is 6C6 * 9C5 = 1 * 126 = 126 ways.Step 6: Add the three cases. Total valid teams = 540 + 504 + 126 = 1170.

Verification / Alternative check:

Why Other Options Are Wrong:

• 1200 and 720: These values do not match the sum of any correct combination of cases and likely come from incorrect use of combinations or ignoring some cases.
• 360: This is much too small and might arise if someone only counted one of the valid cases, such as 6C4 * 9C7 but miscomputed it.

Common Pitfalls:
Students sometimes forget that at least 4 bowlers includes 5 and 6 bowlers, not just exactly 4, leading to undercounting. Another common mistake is to ignore that the total must be 11 and incorrectly pair incorrect numbers of batsmen with a given number of bowlers. It is also easy to mix up when to add and when to multiply counts, so clearly separating cases is essential.

Final Answer:
The number of different 11 player teams that can be formed with at least 4 bowlers is 1170.