I have an upcoming exam, with 5 questions that may cover 10 topics. Suppose I want to accept a 90% probability of my studied exam questions showing up, how many topics should I study? What is the relevant combinatorial calculation here?
If I study 2 topics, what's the chance of these both showing up in my exam?
Assume that each question is on a different topic.
For your second question: $$\frac{1\cdot 1\cdot\binom{8}{3}}{\binom{10}{5}}\approx 22.2\%.$$
It's a little bit counter-intuitive that if you study 3 topics and you want them all show up the chance decrease.
Edit:
With the help from the comment I just figure it out. For your first question you have to study at least 3 questions so at least one of them will should up and fit your criterion, that is:
$$1-\frac{\binom{7}{5}}{\binom{10}{5}} \approx 91.7\%$$
Bonus:
Prepare $6$, at least $3$ shows up:
Total $-$ (only shows up 1 $+$ only shows up 2) :
$$1-\frac{\Bigl(\binom{4}{4}\cdot\binom{6}{1}\Bigr)+\Bigl(\binom{4}{3}\cdot\binom{6}{2}\Bigr)}{\binom{10}{5}}\approx74\%.$$
I would recommend you to at least study about 6 topics so the probability you pass the exam, assume each scores $20$ and the total is $100$, is about $75\%$.