Problem 4. 10 students choose between 3 schools A, B, C randomly (and independently). For each student, there is a $\frac{1}{2}$ chance of going to A, $\frac{1}{3}$ to B, and $\frac{1}{6}$ to C. Let $X_A$, $X_B$, $X_C$ be the number of students choosing A, B, C respectively. Find the pmf of the random vector ($X_A$, $X_B$, $X_C$). Determine whether $X_A$ and $X_B$ are independent.