In how many ways a team of 11 players can be formed out of 25 players, if 6 out of them are always to be included and 5 are always to be excluded?

Options

A. A. 1800
B. B. 1002
C. C. 2002
D. D. none of these

Answer

To solve the problem of forming a team of 11 players out of 25 players, with the conditions that 6 players must always be included and 5 players must always be excluded, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Players**:
We have a total of 25 players.

2. **Account for Included and Excluded Players**:
- 6 players are always included in the team.
- 5 players are always excluded from the team.

3. **Calculate Remaining Players**:
After including the 6 players and excluding the 5 players, we need to determine how many players are left to choose from:
\[ \text{Remaining Players} = 25 - 6 - 5 = 14 \]
So, we have 14 players left to choose from.

4. **Determine How Many More Players are Needed**:
Since we have already included 6 players, we need to select:
\[ \text{Players Needed} = 11 - 6 = 5 \]
We need to choose 5 more players from the remaining 14 players.

5. **Calculate the Number of Combinations**:
The number of ways to choose 5 players from 14 can be calculated using the combination formula \( nCr \):
\[ \text{Number of Ways} = \binom{14}{5} = \frac{14!}{5!(14-5)!} = \frac{14!}{5! \cdot 9!} \]
This simplifies to:
\[ = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} \]

6. **Calculate the Numerator**:
\[ 14 \times 13 = 182 \]
\[ 182 \times 12 = 2184 \]
\[ 2184 \times 11 = 24024 \]
\[ 24024 \times 10 = 240240 \]

7. **Calculate the Denominator**:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

8. **Final Calculation**:
Now divide the numerator by the denominator:
\[ \frac{240240}{120} = 2002 \]

### Final Answer:
Thus, the number of ways to form the team of 11 players is **2002**.

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