Question

A college has 10 basketball players. A 5-member team and a captain will be selected out of these 10 players. How many different selections can be made?

• A. 1260
• B. 1400
• C. 1250
• D. 1600

Hint:

When a problem involves choosing a group *and* assigning a role (like captain) within that group, remember that "choosing 5 and then 1 captain from the 5" is equivalent to "choosing 1 captain from 10 and then 4 members from the remaining 9". Pick the method that seems simpler to calculate for your exam.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem involves selecting a team and a captain from a group of players. This is a problem of combinations and permutations.

Step 2: Key Formula or Approach:
The selection can be broken down into two parts:
1. Choose the 5 members for the team from the 10 players. This is a combination problem: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
2. Choose a captain from the 5 selected members.
Alternatively, we can first choose a captain from the 10 players, and then select the remaining 4 members from the remaining 9 players. This is equivalent.

Step 3: Detailed Explanation:
Method 1: Choose team first, then captain
1. Select 5 players for the team from 10 players:
Number of ways = \( \binom{10}{5} = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} \)
\( = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} \)
\( = \frac{30240}{120} = 252 \)
2. Select 1 captain from the 5 chosen team members:
Number of ways = \( \binom{5}{1} = 5 \)
3. Total different selections:
Total = (Ways to choose team) \(\times\) (Ways to choose captain)
Total = $252 \times 5 = 1260$.
Method 2: Choose captain first, then remaining team members
1. Select 1 captain from 10 players:
Number of ways = \( \binom{10}{1} = 10 \)
2. Select the remaining 4 members from the remaining 9 players:
Number of ways = \( \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4!5!} \)
\( = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} \)
\( = \frac{3024}{24} = 126 \)
3. Total different selections:
Total = (Ways to choose captain) \(\times\) (Ways to choose remaining members)
Total = $10 \times 126 = 1260$.
Both methods yield the same result.

Step 4: Final Answer:
1260 different selections can be made.