Multinomial Covariance and Conditional Distribution

A fair die is rolled $n = 60$ times. Let $X_i$ be the number of times face $i$ appears, for $i = 1, \ldots, 6$, so $(X_1, \ldots, X_6) \sim \text{Multinomial}(60,\, 1/6, \ldots, 1/6)$.

(a) Using indicator variables, compute $\text{Cov}(X_1, X_2)$.

(b) Find the correlation $\rho(X_1, X_2)$.

(c) Determine the conditional distribution of $(X_2, X_3, X_4, X_5, X_6)$ given $X_1 = 12$. What is $E[X_2 \mid X_1 = 12]$?