> For a) This can be done by choosing first the graduate in 4C1 ways and then a UG student in 12C3 ways so in total 4 \* 12C3 ??

Yes!

>For b) I did (4 \* 12C3) / 16C4 , is it correct?

That's for one group. The others should have exactly one graduate too. If you have 4C1 × 12C3 choices for the first group, how many do you have (left) for the second group?

>Or is it 4/4 \* 3/4 \* 2/5 \* 1/4 ?

Are you assigning graduates to groups? After one graduate joins a group, this group has less remaining spots. So the next graduate has a higher chance than 3/4 of joining one of the other three groups.