5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.

1. Find the probability generating function of \(X\).
   Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
   

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   \includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_443_33_675_2013}

   \includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_440_33_1247_2013}

   \includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_441_33_1816_2013}

   \includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_443_31_2385_2015}

3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).