A college has 10 basketball players. In how many different ways can a playing team of 5 members and a separate captain be selected from these 10 players?

Introduction / Context:
This problem involves selecting a team and then designating a captain from among the selected players. It combines combinations and a simple additional choice. It is a standard example of hierarchical selections, where you pick a subset and then pick a special role from within that subset. Such questions are common in aptitude exams and help to clarify the difference between selecting people and assigning roles.

Given Data / Assumptions:
Total number of basketball players = 10. We need to select a team of 5 members. One of the 5 selected players must be chosen as captain. All 10 players are distinct. Only one captain is chosen, and every team has exactly one captain.

Concept / Approach:
First, we choose 5 players out of 10 to form the team, where order does not matter. This is a combination 10C5. After the team is chosen, we then pick one of the 5 team members to be the captain. This is an additional factor of 5. Since these two stages are independent, we multiply the counts. This is an application of the rule of product in combinatorics: total ways = 10C5 * 5.

Step-by-Step Solution:
Step 1: Compute the number of ways to choose 5 players from 10: 10C5. Step 2: 10C5 = 10! / (5! * 5!) = 252. Step 3: For each chosen group of 5 players, any one of the 5 can be chosen as captain. Step 4: Number of ways to choose the captain from the 5 is 5. Step 5: Multiply the two counts: total selections = 10C5 * 5 = 252 * 5. Step 6: Calculate 252 * 5 = 1260. Step 7: Therefore, there are 1260 different ways to form the team and select the captain.

Verification / Alternative check:
You can reverse the steps and first pick the captain and then the remaining 4 players. First, select the captain from the 10 players in 10 ways. After that, choose 4 more players from the remaining 9 to complete the team of 5. The count is 10 * 9C4. Now 9C4 = 126, so total = 10 * 126 = 1260, which matches the earlier method. The equality 10C5 * 5 = 10 * 9C4 is a nice consistency check and confirms our answer.

Why Other Options Are Wrong:
Values like 1400, 1250 or 1600 are not supported by correct combinatorial reasoning. They might come from using permutations directly or from arithmetic errors such as mistakenly computing 10P5 or mixing up factorial values. Only 1260 arises from both valid methods described above and correctly respects the hierarchy of selecting a team and then a captain.

Common Pitfalls:
A common error is to treat the problem as if we are only selecting a captain and four other players without recognizing that the captain must come from within the team. Another mistake is using permutations instead of combinations when forming the team, which overcounts arrangements where the same set of players appears in different orders. Some students also forget the final multiplication by 5 when selecting the captain. Always separate selections (combinations) from role assignments (additional choices) clearly.

Final Answer:
The number of ways to select a 5 member team and a captain from 10 players is 1260.