Question

gebra i b-gr
unit test
unit test complete
time remainn
01:45:10
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 number of ways to choose $k$ items from $n$ items is given by the combination formula:
$$C(n,k)=\frac{n!}{k!(n-k)!}$$

Step2: Calculate Brent's combinations

Brent's club has $n=12$ members, choosing $k=6$:
$$C(12,6)=\frac{12!}{6!(12-6)!}=\frac{12\times11\times10\times9\times8\times7}{6\times5\times4\times3\times2\times1}=924$$

Step3: Calculate Miguel's combinations

Miguel's club has $n=10$ members, choosing $k=6$. Note that $C(n,k)=C(n,n-k)$, so $C(10,6)=C(10,4)$:
$$C(10,6)=\frac{10!}{6!(10-6)!}=\frac{10\times9\times8\times7}{4\times3\times2\times1}=210$$

Step4: Compare the two values

$924 > 210$, so Brent's club has more combinations, which is 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.