In 25 cricket players, there are 10 batsmen, 9 bowlers, 4 all-rounders and 2 wicket-keppers. In how many ways can a team of 11 players be selected which includes 5 batsmen, 4 bowlers, 1 all-rounder and 1 wicketkeeper?

Answer

To solve the problem of selecting a cricket team of 11 players from a pool of 25 players (10 batsmen, 9 bowlers, 4 all-rounders, and 2 wicketkeepers) with specific requirements, we can break down the selection process step by step using combinations.

### Step-by-Step Solution:

1. **Identify the Requirements**:
- We need to select:
- 5 batsmen from 10 available batsmen
- 4 bowlers from 9 available bowlers
- 1 all-rounder from 4 available all-rounders
- 1 wicketkeeper from 2 available wicketkeepers

2. **Calculate the Number of Ways to Select Each Group**:
- **Selecting Batsmen**:
The number of ways to select 5 batsmen from 10 is given by the combination formula:
\[ \text{Number of ways to select batsmen} = \binom{10}{5} \]

- **Selecting Bowlers**:
The number of ways to select 4 bowlers from 9 is:
\[ \text{Number of ways to select bowlers} = \binom{9}{4} \]

- **Selecting All-Rounders**:
The number of ways to select 1 all-rounder from 4 is:
\[ \text{Number of ways to select all-rounders} = \binom{4}{1} \]

- **Selecting Wicketkeepers**:
The number of ways to select 1 wicketkeeper from 2 is:
\[ \text{Number of ways to select wicketkeepers} = \binom{2}{1} \]

3. **Combine the Selections**:
Since the selections are independent, we multiply the number of ways to select each group:
\[ \text{Total ways} = \binom{10}{5} \times \binom{9}{4} \times \binom{4}{1} \times \binom{2}{1} \]

4. **Calculate Each Combination**:
- Calculate \(\binom{10}{5}\):
\[ \binom{10}{5} = \frac{10!}{5! \cdot 5!} = 252 \]
- Calculate \(\binom{9}{4}\):
\[ \binom{9}{4} = \frac{9!}{4! \cdot 5!} = 126 \]
- Calculate \(\binom{4}{1}\):
\[ \binom{4}{1} = 4 \]
- Calculate \(\binom{2}{1}\):
\[ \binom{2}{1} = 2 \]

5. **Final Calculation**:
Now, substitute the values back into the total ways formula:
\[ \text{Total ways} = 252 \times 126 \times 4 \times 2 \]
- Calculate:
\[ 252 \times 126 = 31752 \]
\[ 31752 \times 4 = 127008 \]
\[ 127008 \times 2 = 254016 \]

Thus, the total number of ways to select the team is **254016**.