Bernard Boyer

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

suchonos6r

Expert

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 $\left(\genfrac{}{}{0}{}{7}{2}\right)$. Then you need one more from off campus, so $\left(\genfrac{}{}{0}{}{6}{1}\right)$
Step 2
So the number of groups with two on campus members is
$\left(\genfrac{}{}{0}{}{7}{2}\right)\left(\genfrac{}{}{0}{}{6}{1}\right)=126$
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 $\left(\genfrac{}{}{0}{}{5}{2}\right)$. Then you need one from off campus, so $\left(\genfrac{}{}{0}{}{4}{1}\right)$, giving you
$\left(\genfrac{}{}{0}{}{5}{2}\right)\left(\genfrac{}{}{0}{}{4}{1}\right)=40$
$126-40=86$

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