  In this sequel to Part-I, we present a different approach to bounding the
expected number of real zeroes of random polynomials with real independent
identically distributed coefficients or more generally, exchangeable
coefficients. We show that the mean number of real zeroes does not grow faster
than the logarithm of the degree. The main ingredients of our approach are
Descartes' rule of signs and a new anti-concentration inequality for the
symmetric group. This paper can be read independently of part-I in this series.
