In how many ways can we select a cricket eleven from 17 players in which 5 players can bowl?Each cricket team must include. 2 bowlers.

Answer

To solve the problem of selecting a cricket eleven from 17 players, where 5 players can bowl and the team must include at least 2 bowlers, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Total Players and Requirements**:
- We have a total of 17 players.
- Out of these, 5 players are bowlers, and the remaining 12 are non-bowlers.
- We need to select a team of 11 players, which must include at least 2 bowlers.

2. **Calculate Total Selections Without Conditions**:
- The total number of ways to select 11 players from 17 is given by the combination formula:
\[ \binom{17}{11} \]

3. **Calculate Selections with 0 Bowlers**:
- If we select 0 bowlers, we must select all 11 players from the 12 non-bowlers:
\[ \text{Ways to select 0 bowlers} = \binom{5}{0} \times \binom{12}{11} = 1 \times 12 = 12 \]

4. **Calculate Selections with 1 Bowler**:
- If we select 1 bowler, we choose 1 from the 5 bowlers and 10 from the 12 non-bowlers:
\[ \text{Ways to select 1 bowler} = \binom{5}{1} \times \binom{12}{10} = 5 \times 66 = 330 \]

5. **Calculate Total Selections with Less than 2 Bowlers**:
- Now, we sum the selections with 0 bowlers and 1 bowler:
\[ \text{Total selections with less than 2 bowlers} = 12 + 330 = 342 \]

6. **Calculate Total Selections with At Least 2 Bowlers**:
- To find the selections with at least 2 bowlers, subtract the selections with less than 2 bowlers from the total selections:
\[ \text{Total selections with at least 2 bowlers} = \binom{17}{11} - 342 \]

7. **Calculate \(\binom{17}{11}\)**:
- Using the combination formula:
\[ \binom{17}{11} = \frac{17!}{11! \cdot 6!} = 12376 \]

8. **Final Calculation**:
- Now, substituting back:
\[ \text{Total selections with at least 2 bowlers} = 12376 - 342 = 12034 \]

### Final Answer:
The total number of ways to select a cricket eleven from 17 players, ensuring that at least 2 bowlers are included, is **12034**.