What is the number of ways that a cricket team of 11 players can be made out of 15 players?

Options

A. 364
B. 1001
C. 1365
D. 32760

Answer

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

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

Where:
- \( n \) is the total number of items (in this case, players),
- \( r \) is the number of items to choose (the players we want in the team),
- \( ! \) 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).
- \( r = 11 \) (the number of players to select for the team).

2. **Apply the combination formula**:
\[ C(15, 11) = \frac{15!}{11!(15-11)!} \]
Simplifying \( 15 - 11 \) gives us \( 4 \):
\[ C(15, 11) = \frac{15!}{11! \cdot 4!} \]

3. **Calculate the factorials**:
- \( 15! = 15 \times 14 \times 13 \times 12 \times 11! \)
- \( 11! \) cancels out in the numerator and denominator:
\[ C(15, 11) = \frac{15 \times 14 \times 13 \times 12}{4!} \]

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

5. **Substitute back into the equation**:
\[ C(15, 11) = \frac{15 \times 14 \times 13 \times 12}{24} \]

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

7. **Divide by \( 4! \)**:
\[ C(15, 11) = \frac{32760}{24} = 1365 \]

### Final Answer:
The number of ways to form a cricket team of 11 players from 15 players is **1365**.