Question 81
Let $a$, $b$ be integers chosen independently and uniformly at random from the set $\{0,1,2, \ldots, 80\}$. Compute the expected value of the remainder when the binomial coefficient $\binom{a}{b}=\frac{a!}{b!(a-b)!}$ is divided by $3$. (Here $\binom{0}{0}=1$ and $\binom{a}{b}=0$ whenever $a\lt b$.)