The idea of a probability generating function seems to be a difficult one for sixth formers, and this may well be because the formalism is introduced too abruptly. The t or z or whatever letter is used in a p.g.f. has to be explained as having “no physical significance”, to quote from a recent M.A. booklet on sixth form probability and statistics. The following gentler approach I have found to lead to better understanding on the part of ‘lower fliers’, and it has the advantage of drawing attention to an underlying structure.