Question

brent’s after - school game club has 12 members from which a six - member team is created. miguel’s after - school sports club has 10 members from which a six - member team is created. which student’s club has more possible combinations for his six - member team? miguel’s club has more possible team combinations because he has more members on each team than brent does. miguel’s club has more possible team combinations because there are fewer members in his club. brent’s club has more possible team combinations because there are more members to choose from. brent’s club has more possible team combinations because he has fewer members on each team than miguel’s does.

Explanation:

Step1: Recall Combination Formula

The number of combinations of choosing \( r \) items from \( n \) items is given by \( C(n, r)=\frac{n!}{r!(n - r)!} \), where \( n! = n\times(n - 1)\times\cdots\times1 \).

Step2: Calculate Combinations for Brent's Club

For Brent's club, \( n = 12 \) and \( r = 6 \).

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Step3: Calculate Combinations for Miguel's Club

For Miguel's club, \( n = 10 \) and \( r = 6 \).

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Step4: Compare the Two Results

Since \( 924>210 \), Brent's club has more combinations. The reason is that Brent has more members (\( n = 12 \)) to choose from compared to Miguel (\( n = 10 \)).

Answer:

Brent’s club has more possible team combinations because there are more members to choose from. (Corresponding to the third option: Brent’s club has more possible team combinations because there are more members to choose from.)