Generalisation of Sylvester's problem

Let $P$ be a set of $n$ points in the projective space of dimension $d$ with the property that not all the points are contained in an hyperplane and any $d$ points span an hyperplane. Let an ordinary hyperplane of $P$ be an hyperplane passing through exactly $d$ points of $P$. We show that if $d$ is 3, and $n$ is large and even, then there are precisely $n \left\lfloor \frac{n-1}{4} \right\rfloor$ ordinary planes. Indeed, we describe the exact extemisers for this problem. We also find the number of ordinary hyperplanes for small $n$ and $d$, and lower and upper bounds of this number