A school team has eight volleyball players. A five -member team and a captain will be selected out of these eight players. How many different selections can be made ?

Options

A. 224
B. 112
C. 56
D. None of these

Answer

To solve the problem of selecting a five-member volleyball team and a captain from a group of eight players, we can break it down into two main steps: selecting the team members and then selecting the captain.

### Step-by-Step Solution:

**Step 1: Select the 5 Members from 8 Players**

We need to choose 5 players from the 8 available players. The number of ways to choose r members from n players can be calculated using the combination formula:

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

In our case, we have:

- n = 8 (total players)
- r = 5 (players to be selected)

Using the combination formula, we can also express this as:

\[ 8C5 = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} \]

Calculating this step-by-step:

\[ 8C5 = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

So, there are 56 ways to choose 5 players from 8.

**Step 2: Select the Captain from the 5 Selected Members**

Once we have selected the 5 players, we need to choose 1 captain from these 5 players. The number of ways to choose 1 captain from 5 players is simply:

\[ 5C1 = 5 \]

**Step 3: Calculate the Total Selections**

Now, to find the total number of different selections of a 5-member team and a captain, we multiply the number of ways to choose the team members by the number of ways to choose the captain:

\[ \text{Total Selections} = \text{Number of ways to select 5 members} \times \text{Number of ways to select a captain} \]

\[ \text{Total Selections} = 56 \times 5 = 280 \]

### Final Answer:

Thus, the total number of different selections that can be made is **280**.

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