The random variable \(X\) has probability generating function \(\mathrm{G}_{X}(t)\) given by
\(\mathrm{G}_{X}(t)=\operatorname{ct}(1+t)^{5}\)
where \(c\) is a constant.
(a) Find the value of \(c\).

(b) Find the value of \(\mathrm{E}(X)\).

The random variable \(Y\) is the sum of two independent values of \(X\).
(c) Write down the probability generating function of \(Y\) and hence find \(\operatorname{Var}(Y)\).

(d) Find \(\mathrm{P}(Y=5)\).