  Let $S$ be a set of $n$ points in $\mathbb{R}^3$, no three collinear and not
all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the
total number of planes determined is at least $1 + k
\binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2})$. For similar conditions and
sufficiently large $n$, (inspired by the work of P. D. T. A. Elliott in
\cite{Ell67}) we also show that the number of spheres determined by $n$ points
is at least $1+\binom{n-1}{3}-t_3^{orchard}(n-1)$, and this bound is best
possible under its hypothesis. (By $t_3^{orchard}(n)$, we are denoting the
maximum number of three-point lines attainable by a configuration of $n$
points, no four collinear, in the plane, i.e., the classic Orchard Problem.)
New lower bounds are also given for both lines and circles.
