In how many ways can a team of 11 players be selected from 14 players?

Answer

To solve the problem of selecting a team of 11 players from a pool of 14 players, we will use the concept of combinations. The number of ways to choose \( R \) objects from \( N \) objects is given by the formula:

\[ NCR = \frac{N!}{R!(N-R)!} \]

### Step-by-Step Solution:

1. **Identify the values of N and R**:
- Here, \( N = 14 \) (total players) and \( R = 11 \) (players to be selected).

2. **Use the combination formula**:
- We need to calculate \( 14C11 \).
- According to the formula:
\[ 14C11 = \frac{14!}{11!(14-11)!} = \frac{14!}{11! \cdot 3!} \]

3. **Simplify the factorials**:
- We can simplify \( 14! \) as follows:
\[ 14! = 14 \times 13 \times 12 \times 11! \]
- Therefore, substituting back into the equation:
\[ 14C11 = \frac{14 \times 13 \times 12 \times 11!}{11! \cdot 3!} \]

4. **Cancel out the common \( 11! \)**:
- The \( 11! \) in the numerator and denominator cancels out:
\[ 14C11 = \frac{14 \times 13 \times 12}{3!} \]

5. **Calculate \( 3! \)**:
- \( 3! = 3 \times 2 \times 1 = 6 \).

6. **Substitute \( 3! \) back into the equation**:
\[ 14C11 = \frac{14 \times 13 \times 12}{6} \]

7. **Perform the multiplication**:
- First, calculate \( 14 \times 13 \times 12 \):
\[ 14 \times 13 = 182 \]
\[ 182 \times 12 = 2184 \]

8. **Divide by 6**:
\[ 14C11 = \frac{2184}{6} = 364 \]

9. **Final answer**:
- Therefore, the number of ways to select 11 players from 14 players is \( 364 \).

### Summary:
The total number of ways to select a team of 11 players from 14 players is **364**.