A cricket team of 11 players is to be formed 20 players including 6 bowlers and 3 wicket keepers. In how many different ways can a team be formed so that the team contain exactly 2 wicket keepers and atleast 4 bowlers ?

Options

A. 22725
B. 27225
C. 22275
D. NONE (A), (B), ( C )

Answer

To solve the problem of forming a cricket team of 11 players from a pool of 20 players, including 6 bowlers and 3 wicket keepers, with the constraints of having exactly 2 wicket keepers and at least 4 bowlers, we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Identify the Components of the Team:**
- Total players = 20
- Bowlers = 6
- Wicket Keepers = 3
- Other players (not bowlers or wicket keepers) = 20 - 6 - 3 = 11

2. **Determine the Team Composition:**
- We need exactly 2 wicket keepers.
- We need at least 4 bowlers.
- The total team size is 11 players.

This gives us three possible scenarios for the number of bowlers:
- 4 bowlers, 2 wicket keepers, and 5 other players.
- 5 bowlers, 2 wicket keepers, and 4 other players.
- 6 bowlers, 2 wicket keepers, and 3 other players.

3. **Calculate the Number of Ways for Each Scenario:**

**Scenario 1: 4 Bowlers, 2 Wicket Keepers, 5 Other Players**
- Choose 4 bowlers from 6: \( \binom{6}{4} \)
- Choose 2 wicket keepers from 3: \( \binom{3}{2} \)
- Choose 5 other players from 11: \( \binom{11}{5} \)

The total number of ways for this scenario:
\[ \text{Ways}_1 = \binom{6}{4} \times \binom{3}{2} \times \binom{11}{5} \]

**Scenario 2: 5 Bowlers, 2 Wicket Keepers, 4 Other Players**
- Choose 5 bowlers from 6: \( \binom{6}{5} \)
- Choose 2 wicket keepers from 3: \( \binom{3}{2} \)
- Choose 4 other players from 11: \( \binom{11}{4} \)

The total number of ways for this scenario:
\[ \text{Ways}_2 = \binom{6}{5} \times \binom{3}{2} \times \binom{11}{4} \]

**Scenario 3: 6 Bowlers, 2 Wicket Keepers, 3 Other Players**
- Choose 6 bowlers from 6: \( \binom{6}{6} \)
- Choose 2 wicket keepers from 3: \( \binom{3}{2} \)
- Choose 3 other players from 11: \( \binom{11}{3} \)

The total number of ways for this scenario:
\[ \text{Ways}_3 = \binom{6}{6} \times \binom{3}{2} \times \binom{11}{3} \]

4. **Calculate Each Scenario:**
- For Scenario 1:
\[ \text{Ways}_1 = \binom{6}{4} \times \binom{3}{2} \times \binom{11}{5} = 15 \times 3 \times 462 = 20790 \]
- For Scenario 2:
\[ \text{Ways}_2 = \binom{6}{5} \times \binom{3}{2} \times \binom{11}{4} = 6 \times 3 \times 330 = 5940 \]
- For Scenario 3:
\[ \text{Ways}_3 = \binom{6}{6} \times \binom{3}{2} \times \binom{11}{3} = 1 \times 3 \times 165 = 495 \]

5. **Total Number of Ways:**
\[ \text{Total Ways} = \text{Ways}_1 + \text{Ways}_2 + \text{Ways}_3 = 20790 + 5940 + 495 = 27225 \]

### Final Answer:
The total number of different ways to form the cricket team is **27225**.