  We show that $\scriptstyle{#9(S^2\times S^3)}$ admits an 8-dimensional
complex family of inequivalent non-regular Sasakian-Einstein structures. These
are the first known Einstein metrics on this 5-manifold. In particular, the
bound $\scriptstyle{b_2(M)\leq8}$ which holds for any regular Sasakian-Einstein
$\scriptstyle{M}$ does not apply to the non-regular case. We also discuss the
failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and
describe the orbifold version.
