I uploaded a picture so that my question may be accurate. It is question 3a that I am struggling with. I've learnt that the P(X=r)is the rth derivative(w.r.t. t) of the pgf at t=0 . Divided by r factorial. But I am struggling to figure out how to obtain the rth derivative of this. Should I even try get the rth derivative or is there possibly an easier way?
$$S(t)=\frac8{27}t^3\left(1-\frac{t}3\right)^{-3}=\frac8{27}t^3\sum_{n=0}^\infty \ldots\,\left(\frac{t}3\right)^n=\sum_{n=3}^\infty \ldots\,t^n$$