Robust Covariance of Multinomial Counts

A fair six-sided die is rolled $60$ times independently. Let $N_1$ be the number of times face $1$ appears and $N_2$ the number of times face $2$ appears.

(a) Find $\operatorname{Cov}(N_1, N_2)$.

(b) Use your answer to compute $\operatorname{Var}(N_1 + N_2)$ and verify it by recognizing the distribution of $N_1 + N_2$.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.