There are 15 playears in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 more wicketkeepers. The number of ways, a team of 11 players be selected from them so as to inculede at least 4 bowlers. 5 batsmen and I wicketkeeper is ______ .

Options

A. 888
B. 555
C. 222
D. 777

Answer

To solve the problem of selecting a cricket team from 15 players, we need to ensure that the team includes at least 4 bowlers, 5 batsmen, and 1 wicketkeeper. We can break this down into cases based on the number of bowlers selected.

### Step-by-Step Solution:

1. **Identify the Total Players**:
- Total players = 15
- Bowlers = 6
- Batsmen = 7
- Wicketkeepers = 2

2. **Determine the Team Composition**:
- We need to select a team of 11 players with at least:
- 4 bowlers
- 5 batsmen
- 1 wicketkeeper

3. **Case Analysis**:
- Since we need to have at least 4 bowlers, we can analyze the following cases:
- **Case 1**: 4 bowlers, 6 batsmen, 1 wicketkeeper
- **Case 2**: 4 bowlers, 5 batsmen, 2 wicketkeepers
- **Case 3**: 5 bowlers, 5 batsmen, 1 wicketkeeper
- **Case 4**: 5 bowlers, 4 batsmen, 2 wicketkeepers

4. **Calculating Each Case**:

**Case 1**: 4 bowlers, 6 batsmen, 1 wicketkeeper
- Number of ways to choose 4 bowlers from 6: \( \binom{6}{4} \)
- Number of ways to choose 6 batsmen from 7: \( \binom{7}{6} \)
- Number of ways to choose 1 wicketkeeper from 2: \( \binom{2}{1} \)
- Total for Case 1:
\[ \binom{6}{4} \times \binom{7}{6} \times \binom{2}{1} = 15 \times 7 \times 2 = 210 \]

**Case 2**: 4 bowlers, 5 batsmen, 2 wicketkeepers
- Number of ways to choose 4 bowlers from 6: \( \binom{6}{4} \)
- Number of ways to choose 5 batsmen from 7: \( \binom{7}{5} \)
- Number of ways to choose 2 wicketkeepers from 2: \( \binom{2}{2} \)
- Total for Case 2:
\[ \binom{6}{4} \times \binom{7}{5} \times \binom{2}{2} = 15 \times 21 \times 1 = 315 \]

**Case 3**: 5 bowlers, 5 batsmen, 1 wicketkeeper
- Number of ways to choose 5 bowlers from 6: \( \binom{6}{5} \)
- Number of ways to choose 5 batsmen from 7: \( \binom{7}{5} \)
- Number of ways to choose 1 wicketkeeper from 2: \( \binom{2}{1} \)
- Total for Case 3:
\[ \binom{6}{5} \times \binom{7}{5} \times \binom{2}{1} = 6 \times 21 \times 2 = 252 \]

5. **Total Number of Ways**:
- Add the totals from all cases:
\[ 210 + 315 + 252 = 777 \]

### Final Answer:
The total number of ways to select the team is **777**.