How many days a week you'll be in with vary from course to course, from year to year, and between one semester and the next. And perhaps this will change from one week to the next -- perhaps if you have something that happens in alternate weeks. No one can tell you now what things will look like for you in the future.
It might be that students on your course are in 4 days a week this year, but it might be different next year. So really you have to work on the basis that you'll be in every day and count it as a bonus if some days you don't need to go in.
You'll need to factor in the costs of owning a car (road tax, MOT, insurance,servicing, maintenance, etc) and driving (fuel costs, etc). Google Maps indicates that "Birmingham" to "Nottingham" is around 50 miles, so that's 100 miles a day when you count both directions. And obviously the distance and time will depend on where exactly you live in Birmingham and what the traffic is like at the time you need to travel.
You'll also need to factor in finding somewhere to park in Nottingham, bearing in mind that the university might not provide parking for students, plus whatever the daily cost is.
It looks like there's a significant difference in price between "walk-up" ticket pricing and
Advance tickets, with Advance tickets being significantly cheaper. (But bear in mind that an Advance ticket requires you to travel on the specific train you booked; you can't travel on just any train -- so you're trading flexibility against price.)
You can also save by getting a
16-25 Railcard which will save you one-third on the cost of most fares; but be sure to understand the restrictions that apply.