In how many ways a hockey team of eleven can be elected from 16 players?

Options

A. 4368
B. 4267
C. 5368
D. 4166

Answer

To find the number of ways to select a hockey team of 11 players from a total of 16 players, we can use the combination formula, which 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 (in this case, players to be selected),
- \( ! \) 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, \( n = 16 \) (total players) and \( r = 11 \) (players to be selected).

2. **Apply the combination formula**:
\[ 16C11 = \frac{16!}{11!(16-11)!} = \frac{16!}{11! \cdot 5!} \]

3. **Simplify the factorials**:
- We can expand \( 16! \) as follows:
\[ 16! = 16 \times 15 \times 14 \times 13 \times 12 \times 11! \]
- Thus, we can rewrite the combination as:
\[ 16C11 = \frac{16 \times 15 \times 14 \times 13 \times 12 \times 11!}{11! \cdot 5!} \]

4. **Cancel out \( 11! \)**:
- The \( 11! \) in the numerator and denominator cancels out:
\[ 16C11 = \frac{16 \times 15 \times 14 \times 13 \times 12}{5!} \]

5. **Calculate \( 5! \)**:
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

6. **Substitute \( 5! \) back into the equation**:
\[ 16C11 = \frac{16 \times 15 \times 14 \times 13 \times 12}{120} \]

7. **Calculate the numerator**:
- First, calculate \( 16 \times 15 = 240 \)
- Then, \( 240 \times 14 = 3360 \)
- Next, \( 3360 \times 13 = 43680 \)
- Finally, \( 43680 \times 12 = 524160 \)

8. **Divide by \( 120 \)**:
\[ 16C11 = \frac{524160}{120} = 4368 \]

### Conclusion:
The number of ways to select a hockey team of 11 players from 16 players is **4368**.