There are 17 cricket players, out of which 5 players can bowl. In how many ways can a team of 11 players be selected to as to include 3 bowlers?

Options

A. C( 17,11)
B. C(12,8)
C. C(17,5) `xx` C ( 5,3)
D. C (5,3) `xx` C ( 12,8)

Answer

To solve the problem of selecting a cricket team of 11 players that includes exactly 3 bowlers from a total of 17 players (of which 5 are bowlers), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total players and their categories**:
- Total players = 17
- Bowlers = 5
- Non-bowlers (batsmen) = 17 - 5 = 12

2. **Determine the team composition**:
- We need to select a team of 11 players.
- Out of these, we want to include exactly 3 bowlers.
- Therefore, the number of batsmen in the team will be 11 - 3 = 8 batsmen.

3. **Select the bowlers**:
- We need to choose 3 bowlers from the 5 available bowlers.
- The number of ways to choose 3 bowlers from 5 is given by the combination formula \( C(n, r) \), which is calculated as:
\[ C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

4. **Select the batsmen**:
- We need to choose 8 batsmen from the 12 available batsmen.
- The number of ways to choose 8 batsmen from 12 is given by:
\[ C(12, 8) = C(12, 4) = \frac{12!}{4!(12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

5. **Calculate the total number of ways to form the team**:
- The total number of ways to select the team is the product of the ways to select the bowlers and the batsmen:
\[ \text{Total ways} = C(5, 3) \times C(12, 8) = 10 \times 495 = 4950 \]

### Final Answer:
The total number of ways to select a team of 11 players including exactly 3 bowlers is **4950**.