某大学的篮球俱乐部有 5 名中锋、8 名前锋和 7 名后卫。随着全国学校运动会的临近，教练希望了解俱乐部成员对训练计划的意见。他向俱乐部所有成员发放了问卷，并收到了每个人的回复。

1. State whether it is a sample or a population and explain why based on the context.

The 20 members form a population because the coach wants to find out the opinions of the members of the club, and the 20 members represent the entire group of interest.

1. State an advantage of selecting randomly within each specific category (stratified sampling).

Choosing a random sample in each category guarantees that the team fulfills the positional requirements (exactly 1 center, 2 forwards, and 2 guards). A simple random sample of 5 members drawn from the entire club of 20 might result in an invalid team composition (e.g., picking 3 centers and 2 guards with no forwards). Furthermore, doing so randomly ensures that every member in a given category has an equal, unbiased chance of being selected.

1. Identify the number of ways to choose the players for each position

Number of ways to choose 1 center from 5 = ${}^{5}C_{1}$
Number of ways to choose 2 forwards from 8 = ${}^{8}C_{2}$
Number of ways to choose 2 guards from 7 = ${}^{7}C_{2}$

2. Apply the multiplication principle to find the total number of teams

Total number of teams = ${}^{5}C_{1} \times {}^{8}C_{2} \times {}^{7}C_{2}$
$= 5 \times 28 \times 21$
$= 2940$

1. Determine the number of available players in each specific category after injuries.

Available centers $= 5$
Available pure forwards $= 8 - 1 = 7$
Available pure guards $= 7 - 1 - 1 = 5$
Versatile player (can play F or G) $= 1$

2. Calculate the number of possible teams if the versatile player is selected as a forward, and if selected as a guard.

Case 1: Versatile player is selected as a forward
We need 1 more forward from the 7 pure forwards, and 2 guards from the 5 pure guards.
Number of teams $= \binom{5}{1} \times \binom{7}{1} \times \binom{5}{2} = 5 \times 7 \times 10 = 350$

Case 2: Versatile player is selected as a guard
We need 2 forwards from the 7 pure forwards, and 1 more guard from the 5 pure guards.
Number of teams $= \binom{5}{1} \times \binom{7}{2} \times \binom{5}{1} = 5 \times 21 \times 5 = 525$

3. Calculate the combinations if the versatile player is not selected, and sum the cases for the total.

Case 3: Versatile player is not selected
We need 2 forwards from the 7 pure forwards, and 2 guards from the 5 pure guards.
Number of teams $= \binom{5}{1} \times \binom{7}{2} \times \binom{5}{2} = 5 \times 21 \times 10 = 1050$

Total different teams $= 350 + 525 + 1050 = 1925$