In how many ways can a term of 11 cricketers be chosen from 6 bowlers. 4 wicket keepers and 11 batsmen to give a majority of bastemen if at least 4 bowlers are to be included and there is one wicket keeper?

Options

A. 27730
B. 27720
C. 17720
D. 26720

Answer

To solve the problem of selecting a team of 11 cricketers from 6 bowlers, 4 wicket keepers, and 11 batsmen with the conditions specified, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Requirements**:
- We need a team of 11 players.
- There must be a majority of batsmen.
- At least 4 bowlers must be included.
- Exactly 1 wicket keeper must be included.

2. **Determine Possible Compositions**:
- Since there are 11 players in total and we need a majority of batsmen, the possible distributions of players can be:
- 4 bowlers, 1 wicket keeper, and 6 batsmen (4 + 1 + 6 = 11)
- 5 bowlers, 1 wicket keeper, and 5 batsmen (5 + 1 + 5 = 11)
- 6 bowlers, 1 wicket keeper, and 4 batsmen (6 + 1 + 4 = 11)
- However, the second case (5 bowlers, 1 wicket keeper, and 5 batsmen) does not give a majority to batsmen, so we cannot use it.

3. **Valid Composition**:
- The only valid composition that meets all conditions is:
- 4 bowlers, 1 wicket keeper, and 6 batsmen.

4. **Calculate the Number of Ways to Choose Each Group**:
- **Choosing Bowlers**: We need to choose 4 bowlers from 6 available bowlers.
\[ \text{Ways to choose bowlers} = \binom{6}{4} \]
- **Choosing Wicket Keeper**: We need to choose 1 wicket keeper from 4 available.
\[ \text{Ways to choose wicket keeper} = \binom{4}{1} \]
- **Choosing Batsmen**: We need to choose 6 batsmen from 11 available.
\[ \text{Ways to choose batsmen} = \binom{11}{6} \]

5. **Calculate Each Combination**:
- Calculate \(\binom{6}{4}\):
\[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15 \]
- Calculate \(\binom{4}{1}\):
\[ \binom{4}{1} = 4 \]
- Calculate \(\binom{11}{6}\):
\[ \binom{11}{6} = \frac{11!}{6!(11-6)!} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462 \]

6. **Total Combinations**:
- Multiply the number of ways to choose bowlers, wicket keeper, and batsmen:
\[ \text{Total Ways} = \binom{6}{4} \times \binom{4}{1} \times \binom{11}{6} = 15 \times 4 \times 462 \]
- Calculate:
\[ 15 \times 4 = 60 \]
\[ 60 \times 462 = 27720 \]

### Final Answer:
The total number of ways to choose the team is **27720**.