Problem

Toby has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 3 marbles from the bag, without replacement. The random variable $X$ is the number of red marbles that Toby obtains.

(a) Find the probability generating function of $X$.

Ling also has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 2 marbles from his bag, without replacement. The random variable $Y$ is the number of red marbles that Ling obtains. It is given that the probability generating function of $Y$ is $\frac{1}{12}(1+6t+5t^2)$.

The random variable $Z$ is the total number of red marbles that Toby and Ling obtain.

(b) Find the probability generating function of $Z$, expressing your answer as a polynomial in $t$.

(c) Use the probability generating function of $Z$ to find $\operatorname{Var}(Z)$.