Finding E[XY] to be used in Covariance in a joint probability question

Question

A fair die is rolled $90$ times. Let $X$ be the total number of trials when faces having $2$ or $3$ dots appear and $Y$ be the total number trials when faces having $3$ or $4$ dots appear. Find $\rho(X,Y)$

I found $EX=EY=30$ and $Var(X)=Var(Y)=np(1-p)={90.5\over36} = 25/2$ Now I need to find $E[XY]$ in order to use it in $Cov(X,Y)=E[XY]-E[X]E[X]$ formula. After that I can find $\rho(X,Y)$ but I couldn't find $E[XY]$.

Any help or hint is appreciated.

Answer 1

Consider $W_1$ be the number of times $2$ comes up, $W_2$ be the number of times $3$ comes up, $W_3$ be the number of times $4$ comes up. Then $W_1,W_2,W_3$ are i.i.d. $\operatorname{Binomial}(90,\frac16)$, and $X=W_1+W_2, Y=W_2+W_3$, giving $$E(XY)=E[(W_1+W_2)(W_2+W_3)]$$ The rest is easy due to independence of $W_i$'s.