A Grade 11 Art class is offering students two choices for a project: a pottery project and a mixed media project. Of the $48$ students in the class, $30$ have selected to do the pottery project and $28$ have selected to do the mixed media project (notice some students have decided to do both). If two students are selected at random from the class to show their finished project(s), what is the probability that at least one pottery project and at least one mixed media project will be shown?
So we have $20$ choose to do the media project and $18$ choose to do the mixed media project and $10$ choose to do both. I also know that the denominator is $48C2$. I think I have $$1-\frac{20C2 +18C2}{48C2}.$$ Does this make sense? If so that is $\dfrac{785}{1128}$?
Yes, you are correct. If one project of each type is not shown then this can only be because both chosen students are doing just one project and their projects are the same type.
The probability that the two chosen students are both doing only the pottery project is $\frac{20 \times 19}{48 \times 47}$.
The probability that both are doing only the mixed media project is $\frac{18 \times 17}{48 \times 47}$.
So the probability that at least one project of each type is shown is
$1 - \frac{20 \times 19}{48 \times 47} - \frac{18 \times 17}{48 \times 47} = \frac{1570}{2256} = \frac{785}{1128}$