Alice, Betty, and Carol took the same series of examinations. For each examination there was one mark (grade) of $x$, one mark of $y$, and one mark of $z$; where $x, y, z$ are distinct positive integers. After all the examinations, Alice had a total score of $20$, Betty a total score of $10$, and carol a total score of $9$. If Betty placed first in Algebra, who placed second in Geometry?
What I analyzed(tell me if I'm wrong): There is 3 girls, and they take 3 exams each. If Algebra and Geometry are two of the exams, then should there should be a third subject as well?