Question

A coach needs to select a \(4\)-player starting lineup from a pool of \(10\) players consisting of:
• \(5\) guards
• \(3\) forwards
• \(2\) centres Find the number of different selections if the lineup must include:
• At least \(1\) guard
• At most \(1\) forward
• Exactly \(1\) centre

• A. \(70\)
• B. \(60\)
• C. \(80\)
• D. \(20\)

Hint:

In combination problems with restrictions, divide the problem into separate valid cases and add the results.

Answer 1

The Correct Option is C

Concept: Problems involving team selection are solved using combinations. The combination formula is: \[ {}^nC_r = \frac{n!}{r!(n-r)!} \] Since order does not matter in selecting players, combinations are used instead of permutations.

Step 1: Interpret the conditions carefully. We need a total of \(4\) players. Conditions:
• Exactly \(1\) centre
• At most \(1\) forward
• At least \(1\) guard Available players: \[ 5 \text{ guards},\quad 3 \text{ forwards},\quad 2 \text{ centres} \]

Step 2: Select the centre. Exactly one centre must be selected. Number of ways: \[ {}^2C_1=2 \]

Step 3: Consider forward selections. Since at most one forward is allowed, there are two cases. Case 1: No forward selected. Then remaining players must all be guards. Already chosen: \[ 1 \text{ centre} \] Still need: \[ 3 \text{ guards} \] Ways: \[ {}^5C_3=10 \] Total ways in this case: \[ 2\times10=20 \] Case 2: Exactly one forward selected. Choose: \[ 1 \text{ forward from }3 \] Ways: \[ {}^3C_1=3 \] Already selected: \[ 1 \text{ centre}+1 \text{ forward} \] Need \(2\) more players, both guards. Ways: \[ {}^5C_2=10 \] Total ways in this case: \[ 2\times3\times10 = 60 \]

Step 4: Add both cases. \[ 20+60=80 \] Hence, \[ \boxed{80} \] Therefore the correct answer is: \[ \boxed{(C)\ 80} \]