Question

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

Explanation:

Step1: Recall combination formula

The combination formula is $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n$ is the total number of items and $r$ is the number of items to be chosen.

Step2: Calculate combinations for Brent's club

For Brent's club, $n = 12$ and $r=6$. So $C(12,6)=\frac{12!}{6!(12 - 6)!}=\frac{12!}{6!6!}=\frac{12\times11\times10\times9\times8\times7}{6\times5\times4\times3\times2\times1}=924$.

Step3: Calculate combinations for Miguel's club

For Miguel's club, $n = 10$ and $r = 6$. So $C(10,6)=C(10,4)$ (since $C(n,r)=C(n,n - r)$), and $C(10,4)=\frac{10!}{4!(10 - 4)!}=\frac{10!}{4!6!}=\frac{10\times9\times8\times7}{4\times3\times2\times1}=210$.

Step4: Compare the results

Since $924>210$, Brent's club has more possible combinations because there are more members to choose from.

Answer:

Brent's club has more possible team combinations because there are more members to choose from.