Basic Counting WebWork #1

Hey everyone!

I was having a bit of trouble with one of the questions on the WebWork and was hoping to see if anyone else was having trouble with it and if they figured it out!

The question I am struggling on is:

1. Our Indiscrete Mathematics course has

• 27 students from the the College of Arts, 18 of whom are female;
• 20 students from the the College of Engineering and Informatics, 7 of whom are female;
• 25 students from the the College of Science, 11 of whom are female.

How many ways can we choose three reps , one from each of the three Colleges, so that exactly one is female?

I found the correct answers to the first two sections but I am struggling with the third part. What I did to solve it was:

(18 \cdot 20 \cdot 25) + (27 \cdot 7 \cdot 11) + (27 \cdot 20 \cdot 11) = 19665

I multiplied to find the number of choices of one girl from each college and then added them together to find the whole number of choices of one girl total.

But, WebWork is saying this is the incorrect answer. Does anyone have any ideas?

I think you’ve got the right basic idea, i.e. to break the computation into the sum of three terms - each term representing the number of ways to get three representatives with a female representative from one of the schools. I think each of your terms, though, is too big. Let’s look at the first term, for example:

I guess the 18 is the number of women from the first college, which is good. But, then you multiply by 20, which is the number of potential representatives of either gender from the second college. I think you want to do something to restrict that to just men. Same goes for the 25 as well.