Variance of a die rolls using indicators

Question

I want to find the variance of the number of faces that don't appear when a die is rolled n times. I am thinking of using indicators to find the variance where $I_j$ is the indicator that face j doesn't appear in n rolls. $X=\sum_{j=1}^n I_j$ where $I_j=1$ when face j does not appear in n rolls. Using this I can find that $E(X)= 6(\frac{5}{6})^n$ where $P(I_j=1) = (\frac{5}{6})^n$. The problem is I am not sure how to find $E(X^2)$ for the variance formula since $Var(X) = E(X^2) - (E(X))^2$.

Answer 1

$$X=\sum_{j=1}^6I_j$$ $$X^2=\sum_{j=1}^6I_j^2+\sum_{1\le j\lt k\le6}2I_jI_k$$ $$E(X^2)=6E(I_1^2)+30E(I_1I_2)=6\left(\frac56\right)^n+30\left(\frac46\right)^n$$