A college has 10 basketball players available. A team of 5 players is to be selected, and then one of those 5 players is to be chosen as the captain. In how many different ways can the 5 member team and its captain be selected?

Introduction / Context:
This question is about forming a team and then assigning a special role within that team. We have a pool of 10 distinct players. We first choose a team of 5 players and then select one of those 5 as captain. The team is unordered, but the choice of captain introduces an extra factor because different captains from the same group of players produce different valid outcomes.

Given Data / Assumptions:

• Total available basketball players = 10.
• Team size required = 5 players.
• Exactly one captain must be chosen from among the 5 team members.
• All players are distinct and one person can hold only one role.

Concept / Approach:
This is a two stage process. First, we select the 5 players who will be on the team. This is a combinations problem because the order of players within the team does not matter. Second, from the chosen 5, we select one player to be captain, which is another simple choice. The total number of outcomes is the product of the number of ways to form the team and the number of ways to assign the captaincy within that team.

Step-by-Step Solution:
Step 1: Select 5 players out of 10 to form the team. Number of ways = C(10, 5). C(10, 5) = 252. Step 2: From these 5 team members, choose 1 as captain. Number of ways to choose a captain = 5. Step 3: Multiply to get the total number of team plus captain selections. Total number of outcomes = C(10, 5) * 5 = 252 * 5 = 1260.

Verification / Alternative check:
Another way to see this is to think first of choosing the captain from the 10 players, and then selecting the remaining 4 team members from the remaining 9 players. The number of ways then is 10 * C(9, 4). C(9, 4) = 126, so we get 10 * 126 = 1260, which matches the earlier result. Both approaches yield the same answer, confirming correctness.

Why Other Options Are Wrong:
The value 210 equals C(10, 4) or C(10, 6) and misses the additional factor for choosing the captain. The numbers 720 and 1512 do not match the combination expression for selecting the team and captain. In particular, 720 is 6!, which is unrelated to this setup, and 1512 is larger than the correct count and might result from overcounting some selections or using permutations incorrectly.

Common Pitfalls:
A common error is to use permutations directly on 10 players for 5 positions, which would treat every ordering of the team as different even though the team itself is unordered. Another mistake is to forget that the captain must be one of the 5 selected players and either choose the captain from all 10 and then again from the same set or misuse factorials. Clearly separating the selection of the team from the selection of the captain helps avoid these issues.

Final Answer:
The 5 member team with its captain can be selected in 1260 different ways.