Section B: Probability and Statistics [60 marks] The basketball club in a college has 5 centers, 8 forwards and 7 guards. With the National School Games approaching, the coach wishes to find out the opinions of members of the club about the training programme. He gives a questionnaire to all the members of the club and receives replies from everyone. The coach then decides to select teams to play in the matches for National School Games. A basketball team to play in a match consists of 1 center, 2 forwards and 2 guards.

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:
$\text{Total} = 1050 + 350 + 525 = 1925$