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: Every valid selection sorts the 10 players into exactly three groups: 1 captain, 4 other team members, and 5 players not picked at all.

Step 2: The number of ways to split 10 distinct players into groups of sizes \( 1, 4, 5 \) is given directly by the multinomial coefficient \( \frac{10!}{1! \times 4! \times 5!} \).

Step 3: Computing this: \( \frac{10!}{1! \times 4! \times 5!} = \frac{3628800}{1 \times 24 \times 120} = \frac{3628800}{2880} = 1260 \).
\[ \boxed{1260} \]