If I take a course out of sequence–that is, if I take a course without having taken one or more of its prerequisites, will the course count towards the major?

No. Please note that if the instructor gives you permission to register for a course without having completed the prerequisite, this does not imply that the department will credit the course. In order to take a course out of sequence and have it count towards your major or concentration, you must have written approval from your advisor, and a copy of the approval must be given to the undergraduate coordinator to be put in your student file.