Tanji has a bag containing 4 red balls and 2 blue balls. He selects 3 balls at random from the bag, without replacement. The number of red balls selected by Tanji is denoted by \(X\).
(a) Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

Tanji also has two coins, each biased so that the probability of obtaining a head when it is thrown is \(\frac{1}{4}\). He throws the two coins at the same time. The number of heads obtained is denoted by \(Y\).
(b) Find the probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\).

The random variable \(Z\) is the sum of the number of red balls selected by Tanji and the number of heads obtained.
(c) Find the probability generating function of \(Z\), expressing your answer as a polynomial.

(d) Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\) and \(\operatorname{Var}(Z)\).