If I want to continue mathematics into a PhD program, would doing a bachelor's and master's combined program be a good idea? Would I be required to retake some graduate courses in the PhD program? This would add only one year to my degree from my current uni as opposed to the two from doing a BS and MS separately.
A bit of background information: I am a dual enrollment US student who has completed two years worth of college courses, and so I'll be entering as a freshman with junior standing taking upper division undergrad math courses. Thus, a standalone BS would take me only two years, leaving me to graduate at age 20. Is this too young to start applying to grad school?
Also, I plan to do an undergrad thesis; is one year of familiarizing myself with the faculty too little to choose an adviser? I'm not even sure which math I would want to write about. Will three semesters of intro analysis and intro algebra grad courses be enough to decide on the thesis for my fourth semester?