In how many ways can a cricket team of eleven player be chosen out of a batch of 16 players If a particular player is never chosen then the number of ways in which a cricket team of eleven players can be chosen is :

Options

A. 2365
B. 2359
C. 1365
D. 1056

Answer

To solve the problem of how many ways a cricket team of eleven players can be chosen from a batch of 16 players, given that a particular player is never chosen, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total number of players**:
We have a total of 16 players.

2. **Account for the player who is never chosen**:
Since one particular player is never chosen, we are left with 15 players.

3. **Determine how many players we need to choose**:
We need to select 11 players from the remaining 15 players.

4. **Use the combination formula**:
The number of ways to choose 11 players from 15 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 15 \) and \( r = 11 \).

5. **Plug in the values into the formula**:
\[ \binom{15}{11} = \frac{15!}{11!(15-11)!} = \frac{15!}{11! \cdot 4!} \]

6. **Simplify the factorials**:
\[ \binom{15}{11} = \frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2 \times 1} \]

7. **Calculate the numerator**:
\[ 15 \times 14 = 210 \]
\[ 210 \times 13 = 2730 \]
\[ 2730 \times 12 = 32760 \]

8. **Calculate the denominator**:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

9. **Divide the numerator by the denominator**:
\[ \binom{15}{11} = \frac{32760}{24} = 1365 \]

10. **Conclusion**:
Therefore, the number of ways to choose a cricket team of 11 players from 16 players, with one particular player never chosen, is **1365**.