A cricket team of 11 players is to be formed from 16 players including 4 bowlers and 2 wicketkeepers. In how many different ways can a team be formed so that the team has atleast 3 bowlers and atleast one wicket keeper?

Options

A. a. 2472
B. b. 2274
C. c. 2427
D. d. 1236

Answer

To solve the problem of forming a cricket team of 11 players from a pool of 16 players (which includes 4 bowlers and 2 wicketkeepers) with the conditions of having at least 3 bowlers and at least 1 wicketkeeper, we can break down the solution into several steps.

### Step-by-Step Solution:

1. **Understand the total players and their categories**:
- Total players = 16
- Bowlers = 4
- Wicketkeepers = 2
- Other players = 16 - (4 + 2) = 10

2. **Define the conditions**:
- We need at least 3 bowlers.
- We need at least 1 wicketkeeper.
- The total team size must be 11 players.

3. **Consider the possible combinations of bowlers and wicketkeepers**:
- Since we have 4 bowlers, the possible scenarios for bowlers are:
- 3 bowlers and 1 wicketkeeper
- 3 bowlers and 2 wicketkeepers
- 4 bowlers and 1 wicketkeeper
- 4 bowlers and 2 wicketkeepers

4. **Calculate the number of ways for each scenario**:

**Scenario 1**: 3 bowlers and 1 wicketkeeper
- Choose 3 bowlers from 4: \( \binom{4}{3} = 4 \)
- Choose 1 wicketkeeper from 2: \( \binom{2}{1} = 2 \)
- Choose the remaining 7 players from the other 10 players: \( \binom{10}{7} = \binom{10}{3} = 120 \)
- Total ways for this scenario: \( 4 \times 2 \times 120 = 960 \)

**Scenario 2**: 3 bowlers and 2 wicketkeepers
- Choose 3 bowlers from 4: \( \binom{4}{3} = 4 \)
- Choose 2 wicketkeepers from 2: \( \binom{2}{2} = 1 \)
- Choose the remaining 6 players from the other 10 players: \( \binom{10}{6} = 210 \)
- Total ways for this scenario: \( 4 \times 1 \times 210 = 840 \)

**Scenario 3**: 4 bowlers and 1 wicketkeeper
- Choose 4 bowlers from 4: \( \binom{4}{4} = 1 \)
- Choose 1 wicketkeeper from 2: \( \binom{2}{1} = 2 \)
- Choose the remaining 6 players from the other 10 players: \( \binom{10}{6} = 210 \)
- Total ways for this scenario: \( 1 \times 2 \times 210 = 420 \)

**Scenario 4**: 4 bowlers and 2 wicketkeepers
- Choose 4 bowlers from 4: \( \binom{4}{4} = 1 \)
- Choose 2 wicketkeepers from 2: \( \binom{2}{2} = 1 \)
- Choose the remaining 5 players from the other 10 players: \( \binom{10}{5} = 252 \)
- Total ways for this scenario: \( 1 \times 1 \times 252 = 252 \)

5. **Add all the scenarios together**:
- Total ways = 960 + 840 + 420 + 252 = 2472

### Final Answer:
The total number of ways to form the cricket team under the given conditions is **2472**.