In how many ways can a cricket team of 11 players be chosen out of a batch of 15 players? There is no restriction on the selection.

Options

A. a. 2365
B. b. 1259
C. c. 1365
D. d. 1056

Answer

To find the number of ways to choose a cricket team of 11 players from a batch of 15 players, we can use the concept of combinations. The formula for combinations is given by:

\[ nCr = \frac{n!}{r!(n-r)!} \]

Where:
- \( n \) is the total number of items (in this case, players),
- \( r \) is the number of items to choose (the team size),
- \( ! \) denotes factorial, which is the product of all positive integers up to that number.

### Step-by-Step Solution:

1. **Identify the values of \( n \) and \( r \)**:
- Here, \( n = 15 \) (the total number of players) and \( r = 11 \) (the number of players to choose).

2. **Apply the combination formula**:
\[ 15C11 = \frac{15!}{11!(15-11)!} = \frac{15!}{11! \cdot 4!} \]

3. **Simplify the factorials**:
- We can expand \( 15! \) as:
\[ 15! = 15 \times 14 \times 13 \times 12 \times 11! \]
- Thus, we can rewrite the combination as:
\[ 15C11 = \frac{15 \times 14 \times 13 \times 12 \times 11!}{11! \cdot 4!} \]

4. **Cancel out \( 11! \)**:
\[ 15C11 = \frac{15 \times 14 \times 13 \times 12}{4!} \]

5. **Calculate \( 4! \)**:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

6. **Substitute \( 4! \) back into the equation**:
\[ 15C11 = \frac{15 \times 14 \times 13 \times 12}{24} \]

7. **Calculate the numerator**:
- First, calculate \( 15 \times 14 = 210 \)
- Then, \( 210 \times 13 = 2730 \)
- Finally, \( 2730 \times 12 = 32760 \)

8. **Divide by \( 24 \)**:
\[ 15C11 = \frac{32760}{24} = 1365 \]

Thus, the number of ways to choose a cricket team of 11 players from 15 players is **1365**.

### Final Answer:
The number of ways to choose the team is **1365**.