In how many ways can a cricket team of eleven players be chosen out of a batch of 16 players if a particular player is always chosen?

Options

A. 2002
B. 3003
C. 1003
D. 7603

Answer

To solve the problem of how many ways a cricket team of eleven players can be chosen out of a batch of 16 players, given that a particular player is always chosen, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to select a cricket team of 11 players from a total of 16 players. One specific player is always included in the team.

2. **Fix the Selected Player**: Since one player is always chosen, we can consider this player as already selected. This means we only need to choose the remaining players.

3. **Calculate Remaining Players**: After selecting the fixed player, we have 15 players left (16 total - 1 fixed player = 15 remaining players).

4. **Determine Players to Choose**: Since we have already chosen 1 player, we need to select 10 more players from the remaining 15 players.

5. **Use Combinations Formula**: The number of ways to choose 10 players from 15 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 15 \) and \( r = 10 \).

6. **Calculate the Combination**:
\[ \binom{15}{10} = \frac{15!}{10!(15-10)!} = \frac{15!}{10! \cdot 5!} \]

7. **Simplify the Factorials**:
- Calculate \( 15! \) as \( 15 \times 14 \times 13 \times 12 \times 11 \times 10! \).
- The \( 10! \) cancels out:
\[ \binom{15}{10} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5!} \]

8. **Calculate \( 5! \)**:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

9. **Substitute and Calculate**:
\[ \binom{15}{10} = \frac{15 \times 14 \times 13 \times 12 \times 11}{120} \]

10. **Perform the Multiplication**:
- Calculate \( 15 \times 14 = 210 \)
- Calculate \( 210 \times 13 = 2730 \)
- Calculate \( 2730 \times 12 = 32760 \)
- Calculate \( 32760 \times 11 = 360360 \)

11. **Divide by 120**:
\[ \frac{360360}{120} = 3003 \]

12. **Final Answer**: The total number of ways to choose the cricket team is **3003**.