Jason has three biased coins. For each coin the probability of obtaining a head when it is thrown is \(\frac{2}{3}\). Jason throws all three coins. The number of heads obtained is denoted by \(X\).

(a) Find the probability generating function \(G_X(t)\) of \(X\).

Jason also has two unbiased coins. He throws all five coins. The number of heads obtained from the two unbiased coins is denoted by \(Y\). It is given that \(G_Y(t)=\frac14+\frac12t+\frac14t^2\). The random variable \(Z\) is the total number of heads obtained when Jason throws all five coins.

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

(c) Find \(E(Z)\).