If my University is Public, Can I use?
The 'University' can't use the home version of avast, it must use the Pro version, though if they contact sales (at) avast (dot) com, they may get an educational discount.
If you are a student and this is your laptop that you take into university (from your home) to use in your studies, then you could use that license on a computer in your home and on a laptop that you use for your studies at university.