You're a university student studying Business. You're taking one art class this semester just for fun. But you're only attending the class, not getting a grade for it. Another student in the class finds out that you're studying business, so she asks why you're taking this class. You respond:
I'm just auditing it, so it doesn't count toward my major.
When you take classes, you usually take them for credit. This means that you get a grade and the class counts toward your degree. But in most colleges and universities, you can also "audit" a class. This means that you just come to the class and listen to the lectures, but you don't get a grade and the class doesn't count toward your degree.
When you have a specific goal, something that "counts toward" that goal is officially recognized as a "point" that brings you closer to the goal. In many situations, there are some actions that "count toward" your goal and others that do not. For example, if you're a salesperson you may have a sales target. If you sell to new clients, the amount is added to your total sales amount, but things you sell to previous clients are not added. You would say:
Sales to existing clients don't count toward my weekly sales targets.
Your major is the field that you get a degree in. Example majors are Biology, History, Psychology, Engineering, Computer Science, Business, etc. Students who are meeting each other for the first time often ask:
What's your major?
"Major" is usually just used to talk about a person's undergraduate college or university degree.
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