  We present various results on multiplying cycles in the symmetric group. Our
first result is a generalisation of the following theorem of Boccara (1980):
the number of ways of writing an odd permutation in the symmetric group on $n$
symbols as a product of an $n$-cycle and an $n-1$-cycle is independent of the
permutation chosen. We give a number of different approaches of our
generalisation. One partial proof uses an inductive method which we also apply
to other problems. In particular, we give a formula for the distribution of the
number of cycles over all products of cycles of fixed lengths. Another
application is related to the recent notion of separation probabilities for
permutations introduced by Bernardi, Du, Morales and Stanley (2014).
