A class has 8 football players. A team of 5 members and a separate captain are to be selected from these 8 players (so that there are 6 distinct people: 1 captain and 5 other team members). In how many different ways can such a selection be made?

Introduction / Context:
This selection problem tests the understanding of combinations under an additional role constraint. We not only choose members for a team but also designate one distinct person as captain. It is important to correctly interpret that the captain is not an extra role for a chosen member of the team but a separate person in addition to the five other team members.

Given Data / Assumptions:

• There are 8 distinct football players.
• A team of 5 members is to be chosen.
• A separate captain is also to be chosen from the remaining players, making a total of 6 distinct people (5 members plus 1 captain).
• Order among the 5 team members does not matter, but the captain is a special designated role.

Concept / Approach:
We should choose the captain first or the team first; both methods yield the same count. The most straightforward approach is to first choose the captain from all 8 players, and then choose the remaining 5 team members from the remaining 7 players. Since the captain is not counted among the 5, the selection steps are clearly separated. The total number of ways is the product of the ways of these two choices.

Step-by-Step Solution:
Step 1: Choose the captain from the 8 players. There are 8 possible choices for the captain.Step 2: After selecting the captain, there are 7 players left.Step 3: Choose 5 team members from these remaining 7 players. This can be done in 7C5 ways.Step 4: Compute 7C5 = 7C2 = (7 × 6) / (2 × 1) = 21.Step 5: Total number of ways = 8 (choices for captain) × 21 (choices for team) = 168.
Verification / Alternative check:
An alternative perspective is to first choose the group of 6 people who will be involved (5 members plus the eventual captain). This can be done in 8C6 = 28 ways. From each such selected group of 6, we can then choose 1 of the 6 to be the captain in 6 ways, and the remaining 5 automatically become the team members. Thus total ways would be 8C6 × 6 = 28 × 6 = 168, which matches our previous result.

Why Other Options Are Wrong:
210 would be 7C5 × 6 if misapplied, or 8C4 × something, and does not correctly describe the two stage selection. 1260 and 10!/6! are much larger and not connected to the correct combinatorial structure here. Only 168 correctly counts the combinations consistent with the problem statement.

Common Pitfalls:
The biggest source of error is misinterpreting whether the captain is one of the five team members or an additional separate person. If students assume the captain is chosen from within the team, they will compute 8C5 × 5, which equals 56 × 5 = 280, a different answer. Carefully reading that the team of 5 and the captain are separate roles is essential.

Final Answer:
The number of different selections of a 5 member team and a separate captain is 168.