Correlation between counts of ones and fives

A fair six-sided die is thrown $n$ times. Write $X$ for how many throws come up $1$ and $Y$ for how many throws come up $5$. What is the correlation coefficient $\text{Corr}(X,Y)$?

$-\tfrac15$. Each count is $\text{Bin}(n,\tfrac16)$ with variance $\tfrac{5n}{36}$. On a single throw the two indicators can't both be 1, so each throw contributes covariance $-\tfrac1{36}$, giving $\text{Cov}(X,Y)=-\tfrac{n}{36}$. Dividing by $\tfrac{5n}{36}$ gives $-\tfrac15$, independent of $n$.