brents after - school game club has 12 members from which a six - member team is created. miguels after…

Answer:

We need to use the combination formula (C(n,r)=\frac{n!}{r!(n - r)!}), where (n) is the total number of members in the club and (r) is the number of members in the team.

For Brent's club: (n_1 = 12) and (r_1=6). [ \begin{align*} 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 \end{align*} ]

For Miguel's club: (n_2 = 10) and (r_2 = 6). [ \begin{align*} C(10,6)&=C(10,4)\quad(\text{since }C(n,r)=C(n,n - r))\ &=\frac{10!}{4!(10 - 4)!}\ &=\frac{10!}{4!6!}\ &=\frac{10\times9\times8\times7}{4\times3\times2\times1}\ &=210 \end{align*} ]

Since (924>210), Brent's club has more possible team - combinations because there are more members to choose from. So the answer is: Brent's club has more possible team combinations because there are more members to choose from.

Explanation:

Step1: Recall combination formula

(C(n,r)=\frac{n!}{r!(n - r)!})

Step2: Calculate combinations for Brent's club

(C(12,6)=\frac{12!}{6!6!}=924)

Step3: Calculate combinations for Miguel's club

(C(10,6)=C(10,4)=\frac{10!}{4!6!}=210)

Step4: Compare results

Since (924 > 210), Brent's club has more combinations.