The Indian cricket team has 16 players including 2 wicket keepers and 5 bowlers. In how many ways can a cricket eleven be selected if the team must contain exactly 1 wicket keeper and at least 4 bowlers?

Introduction / Context:
This question tests multiple combination ideas together. The team selection has role based constraints: exactly one of the two wicket keepers must be chosen, and the number of bowlers must be at least 4 out of 5 available bowlers. The remaining players come from the other group of non bowler and non wicket keeper players. We tackle the problem by splitting into cases based on how many bowlers are chosen and then summing the counts.

Given Data / Assumptions:
Total players = 16. Wicket keepers = 2. Bowlers = 5. Other players (neither wicket keeper nor bowler) = 16 - 2 - 5 = 9. The team size is 11 players. Exactly 1 wicket keeper must be selected. At least 4 bowlers must be selected.

Concept / Approach:
We first choose 1 wicket keeper from 2. Then we consider two possible cases for the bowlers: choosing exactly 4 bowlers from 5, or choosing all 5 bowlers. For each case, the remaining players needed to complete 11 are chosen from the 9 other players. We use combinations for each selection and multiply where appropriate. Finally, we sum the counts from the two cases to get the total number of valid teams.

Step-by-Step Solution:
Step 1: Choose the wicket keeper. Ways = 2C1 = 2. Step 2: Case 1: Team has exactly 4 bowlers. Step 3: Choose 4 bowlers from 5: 5C4 = 5. Step 4: So far selected players = 1 wicket keeper + 4 bowlers = 5. Step 5: Remaining players needed = 11 - 5 = 6, chosen from the 9 other players. Step 6: Ways to choose these 6 players = 9C6 = 84. Step 7: Total teams in Case 1 = 2 * 5 * 84 = 2 * 420 = 840. Step 8: Case 2: Team has all 5 bowlers. Step 9: Choose 5 bowlers from 5: 5C5 = 1. Step 10: So far selected players = 1 wicket keeper + 5 bowlers = 6. Step 11: Remaining players needed = 11 - 6 = 5, chosen from the 9 other players. Step 12: Ways to choose these 5 players = 9C5 = 126. Step 13: Total teams in Case 2 = 2 * 1 * 126 = 252. Step 14: Total valid teams = Case 1 + Case 2 = 840 + 252 = 1092.

Verification / Alternative check:
You can verify the combination values individually: 9C6 = 84 and 9C5 = 126 can be checked using factorial expressions or known combinatorial symmetry 9C6 = 9C3 and 9C5 = 9C4. Also, the sum of bowlers in all teams should logically range only over 4 or 5, which we have fully covered. There is no overlap between Case 1 and Case 2 because the number of bowlers is different. Therefore, adding the counts is valid, and 1092 is consistent.

Why Other Options Are Wrong:
Values like 1024, 1900 or 2000 arise from incorrect combinations or forgetting to split into cases for the number of bowlers. For example, counting teams with no restriction on bowlers or using 16C11 directly would ignore the specific constraints and give a completely different value. Only 1092 follows logically from selecting the wicket keeper, then the bowlers, then the remaining players under the given roles.

Common Pitfalls:
A common mistake is to ignore the requirement of exactly one wicket keeper and at least four bowlers, leading to overcounting or undercounting. Some students treat at least 4 bowlers as exactly 4 bowlers and forget Case 2 where all 5 bowlers are in the team. Another mistake is to choose players in one big step without using separate combinations for each role, which often causes miscounting. Breaking the problem into clear cases and applying the combination formula carefully avoids these errors.

Final Answer:
The cricket eleven can be selected in 1092 different ways under the given conditions.