Data and Displays · Challenge problem

A frequency table of quiz scores reads: score $1$: $2$ students; score $2$: $a$; score $3$: $5$; score $4$: $b$; score $5$: $3$, where $a$ and $b$ are positive whole numbers. The mean score is exactly $3.2$ and the median is exactly $3$. What is the largest possible number of students?