In how many ways, a cricket team of 11 players can be made from 15 players, if a particular player is never chosen?

Options

A. 364
B. 480
C. 1365
D. 640

Answer

To solve the problem of how many ways a cricket team of 11 players can be formed from 15 players when a particular player is never chosen, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Players**:
We start with a total of 15 players.

2. **Exclude the Particular Player**:
Since one specific player is never chosen, we exclude this player from our selection. This leaves us with:
\[ 15 - 1 = 14 \text{ players} \]

3. **Determine the Team Size**:
We need to select a team of 11 players from the remaining 14 players.

4. **Use the Combination Formula**:
The number of ways to choose 11 players from 14 can be calculated using the combination formula:
\[ nCr = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, and \( ! \) denotes factorial.

5. **Apply the Formula**:
Here, \( n = 14 \) and \( r = 11 \):
\[ 14C11 = \frac{14!}{11!(14-11)!} = \frac{14!}{11! \cdot 3!} \]

6. **Simplify the Factorials**:
We can simplify this expression:
\[ 14C11 = \frac{14 \times 13 \times 12 \times 11!}{11! \times 3!} \]
The \( 11! \) in the numerator and denominator cancels out:
\[ = \frac{14 \times 13 \times 12}{3!} \]

7. **Calculate \( 3! \)**:
The value of \( 3! \) is:
\[ 3! = 3 \times 2 \times 1 = 6 \]

8. **Substitute and Calculate**:
Now substitute \( 3! \) back into the equation:
\[ = \frac{14 \times 13 \times 12}{6} \]

9. **Perform the Multiplication**:
Calculate \( 14 \times 13 \times 12 \):
\[ 14 \times 13 = 182 \]
\[ 182 \times 12 = 2184 \]

10. **Divide by 6**:
Now divide by 6:
\[ \frac{2184}{6} = 364 \]

### Final Answer:
The number of ways to form a cricket team of 11 players from 15 players, excluding one particular player, is **364**.