B 组：概率与统计 [60 分] 某大专的篮球俱乐部有 5 名中锋、8 名前锋和 7 名后卫。随着全国学校运动会（National School Games）的临近，教练希望了解俱乐部成员对训练计划的意见。他向俱乐部的所有成员发放了问卷，并收到了每个人的回复。 教练随后决定选拔参赛队伍参加全国学校运动会的比赛。参加比赛的篮球队由 1 名中锋、2 名前锋和 2 名后卫组成。

1. State whether the group is a population or a sample, and provide the justification based on the context.

The coach wants to find out the opinions of the "members of the club". The club has $5 + 8 + 7 = 20$ members. Since the 20 members constitute the entire target group of interest for the survey, they form a population.

1. State the advantage of category-based (stratified) random sampling over simple random sampling for this specific context.

Choosing a random sample in each category ensures that the team formed will have the exact required composition of playing positions (1 center, 2 forwards, and 2 guards) for the match. A simple random sample of 5 members from the entire club might fail to produce a valid team composition.

1. Set up the product of combinations for each position

$\text{Number of teams} = {}^{5}C_1 \times {}^{8}C_2 \times {}^{7}C_2$

2. Evaluate the combinations to find the total number of teams

$= 5 \times 28 \times 21 = 2940$

1. Identify the available players and their specific roles

After the 2 injuries, the available players are:

• $5$ Centers
• $7$ Forwards (strictly forwards)
• $6$ Guards, comprising $5$ strictly Guards and $1$ versatile player (who can play as a Forward or Guard).

2. Calculate the number of teams if the versatile player is NOT selected

Choose $1$ Center from $5$, $2$ Forwards from $7$, and $2$ Guards from $5$:
$\text{Ways} = {^5C_1} \times {^7C_2} \times {^5C_2} = 5 \times 21 \times 10 = 1050$

3. Calculate the number of teams if the versatile player is selected as a Forward

The versatile player fills $1$ Forward spot. We need $1$ Center from $5$, $1$ more Forward from $7$, and $2$ Guards from $5$:
$\text{Ways} = {^5C_1} \times {^7C_1} \times {^5C_2} = 5 \times 7 \times 10 = 350$

4. Calculate the number of teams if the versatile player is selected as a Guard, and find the overall total

The versatile player fills $1$ Guard spot. We need $1$ Center from $5$, $2$ Forwards from $7$, and $1$ more Guard from $5$:
$\text{Ways} = {^5C_1} \times {^7C_2} \times {^5C_1} = 5 \times 21 \times 5 = 525$

Total number of teams is the sum of these mutually exclusive cases:
$\text{Total} = 1050 + 350 + 525 = 1925$