Bernard Boyer

Answered

2022-07-15

Statistics with discrete math

An upper-level math class has 13 students: 4 of them are females. Two of the females and four of the males live oﬀ campus. How many ways can a project group of three students be chosen so that it has two on-campus members and at least one female? Explain your answer

An upper-level math class has 13 students: 4 of them are females. Two of the females and four of the males live oﬀ campus. How many ways can a project group of three students be chosen so that it has two on-campus members and at least one female? Explain your answer

Answer & Explanation

suchonos6r

Expert

2022-07-16Added 14 answers

Step 1

13 students composed of 9 males, 4 females.

4 females composed of 2 on campus, 2 off campus.

9 males composed of 5 on campus, 4 off campus.

13 students composes of 7 on campus, 6 off campus.

Let's find the amount of groups that have two on campus members.

You need to pick 2 of the 7, so $(}\genfrac{}{}{0ex}{}{7}{2}{\textstyle )$. Then you need one more from off campus, so $(}\genfrac{}{}{0ex}{}{6}{1}{\textstyle )$

Step 2

So the number of groups with two on campus members is

$(}\genfrac{}{}{0ex}{}{7}{2}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{6}{1}{\textstyle )}=126$

Now you need at least one female. So let's find all of the groups of 2 on campus members and no females, then eliminate that from the above group.

You need to pick 2 on campus males. There are 5 to choose from, so $(}\genfrac{}{}{0ex}{}{5}{2}{\textstyle )$. Then you need one from off campus, so $(}\genfrac{}{}{0ex}{}{4}{1}{\textstyle )$, giving you

$(}\genfrac{}{}{0ex}{}{5}{2}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{4}{1}{\textstyle )}=40$

$126-40=86$

13 students composed of 9 males, 4 females.

4 females composed of 2 on campus, 2 off campus.

9 males composed of 5 on campus, 4 off campus.

13 students composes of 7 on campus, 6 off campus.

Let's find the amount of groups that have two on campus members.

You need to pick 2 of the 7, so $(}\genfrac{}{}{0ex}{}{7}{2}{\textstyle )$. Then you need one more from off campus, so $(}\genfrac{}{}{0ex}{}{6}{1}{\textstyle )$

Step 2

So the number of groups with two on campus members is

$(}\genfrac{}{}{0ex}{}{7}{2}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{6}{1}{\textstyle )}=126$

Now you need at least one female. So let's find all of the groups of 2 on campus members and no females, then eliminate that from the above group.

You need to pick 2 on campus males. There are 5 to choose from, so $(}\genfrac{}{}{0ex}{}{5}{2}{\textstyle )$. Then you need one from off campus, so $(}\genfrac{}{}{0ex}{}{4}{1}{\textstyle )$, giving you

$(}\genfrac{}{}{0ex}{}{5}{2}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{4}{1}{\textstyle )}=40$

$126-40=86$

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