Out of 16 players of a cricket team, 4 are bowlers and 2 are wicket keepers. A team of 11 players is to be chosen so as to contain at least 3 bowlers and at least 1 wicket keeper. The number of ways in which the team be selected is

Options

A. 2400
B. 2472
C. 2500
D. 960

Answer

To solve the problem of selecting a cricket team of 11 players from a total of 16 players (4 bowlers and 2 wicket keepers), ensuring that the team contains at least 3 bowlers and at least 1 wicket keeper, we can break it down into cases based on the number of bowlers and wicket keepers selected.

### Step-by-Step Solution:

1. **Identify Total Players and Requirements**:
- Total players = 16
- Bowlers = 4
- Wicket Keepers = 2
- Team size = 11
- Conditions: At least 3 bowlers and at least 1 wicket keeper.

2. **Case 1: 3 Bowlers and 1 Wicket Keeper**:
- Choose 3 bowlers from 4: \( \binom{4}{3} \)
- Choose 1 wicket keeper from 2: \( \binom{2}{1} \)
- Remaining players to choose = 11 - (3 + 1) = 7 from the remaining players (16 - 4 - 2 = 10).
- Total ways for this case:
\[ \binom{4}{3} \times \binom{2}{1} \times \binom{10}{7} = 4 \times 2 \times 120 = 960 \]

3. **Case 2: 3 Bowlers and 2 Wicket Keepers**:
- Choose 3 bowlers from 4: \( \binom{4}{3} \)
- Choose 2 wicket keepers from 2: \( \binom{2}{2} \)
- Remaining players to choose = 11 - (3 + 2) = 6 from the remaining players (10).
- Total ways for this case:
\[ \binom{4}{3} \times \binom{2}{2} \times \binom{10}{6} = 4 \times 1 \times 210 = 840 \]

4. **Case 3: 4 Bowlers and 1 Wicket Keeper**:
- Choose 4 bowlers from 4: \( \binom{4}{4} \)
- Choose 1 wicket keeper from 2: \( \binom{2}{1} \)
- Remaining players to choose = 11 - (4 + 1) = 6 from the remaining players (10).
- Total ways for this case:
\[ \binom{4}{4} \times \binom{2}{1} \times \binom{10}{6} = 1 \times 2 \times 210 = 420 \]

5. **Case 4: 4 Bowlers and 2 Wicket Keepers**:
- Choose 4 bowlers from 4: \( \binom{4}{4} \)
- Choose 2 wicket keepers from 2: \( \binom{2}{2} \)
- Remaining players to choose = 11 - (4 + 2) = 5 from the remaining players (10).
- Total ways for this case:
\[ \binom{4}{4} \times \binom{2}{2} \times \binom{10}{5} = 1 \times 1 \times 252 = 252 \]

6. **Total Ways to Select the Team**:
- Sum of all cases:
\[ 960 + 840 + 420 + 252 = 2472 \]

### Final Answer:
The total number of ways to select the team is **2472**.