A team of 8 players is to be chosen from a group of 12 players. Out of the eight players one is to be elected as captain and another vice-captain . In how many ways can this be done?

Options

A. 27720
B. 13860
C. 6930
D. 495

Answer

To solve the problem of selecting a team of 8 players from a group of 12 players, and then electing a captain and a vice-captain from those 8 players, we can break the solution down into clear steps.

### Step-by-Step Solution:

1. **Choosing the Team of 8 Players from 12 Players:**
We need to select 8 players from a total of 12 players. The number of ways to choose 8 players from 12 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 12 \) and \( r = 8 \):
\[ \binom{12}{8} = \frac{12!}{8!(12-8)!} = \frac{12!}{8! \cdot 4!} \]

2. **Calculating the Factorial Values:**
We can simplify this:
\[ \binom{12}{8} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} \]
Calculate the numerator:
\[ 12 \times 11 = 132, \quad 132 \times 10 = 1320, \quad 1320 \times 9 = 11880 \]
Now, calculate the denominator:
\[ 4 \times 3 \times 2 \times 1 = 24 \]
Now divide:
\[ \frac{11880}{24} = 495 \]
So, there are 495 ways to choose 8 players from 12.

3. **Choosing the Captain and Vice-Captain:**
After selecting 8 players, we need to choose a captain and a vice-captain. The captain can be chosen in 8 ways (from the 8 players), and once the captain is chosen, there are 7 remaining players from which to choose the vice-captain. Therefore, the number of ways to choose the captain and vice-captain is:
\[ 8 \times 7 = 56 \]

4. **Calculating the Total Number of Ways:**
Finally, we multiply the number of ways to choose the team by the number of ways to choose the captain and vice-captain:
\[ \text{Total Ways} = 495 \times 56 \]
Now calculate:
\[ 495 \times 56 = 27720 \]

Thus, the total number of ways to choose the team of 8 players and elect a captain and vice-captain is **27720**.

### Final Answer:
The total number of ways to choose a team of 8 players from 12 and elect a captain and vice-captain is **27720**.

---