  Let $T\subset \R^{m+1}$ be a strictly convex domain bounded by a smooth
hypersurface $X=\partial T$. In this paper we find lower bounds on the number
of billiard trajectories in $T$ which have a prescribed intial point $A\in X$,
a prescribed final point $B\in X$ and make a prescribed number $n$ of
reflections at the boundary $X$. We apply a topological approach based on
calculation of cohomology rings of certain configuration spaces.
